Cryptic crosswords have complex, two-part clues. British-style crosswords have looser rules behind their clues, but in the U. S. they were adapted into cryptic crosswords, which follow the rules outlined below fairly strictly.

1. Each clue consists of two separate parts, the straight definition and the wordplay, run together to make a single statement that usually makes some sort of sense (or nonsense), called the surface reading. The wordplay may be any of the types described in the following rules. These two parts may be separated by a connective word or run directly together, but they do not overlap or intermingle; the straight definition will always appear at the beginning or end of the clue. The one exception to this rule is &lit., or literal clues. In these clues, the entire clue is a straight definition (though perhaps a stilted or punny one), and the entire clue can be read differently to be a wordplay clue for the answer. The classic example is "A grim era, perhaps!" (solution). These clues are customarily denoted by an exclamation point (!) at the end of the clue. Watch out for clues where words in either part of the clue have a different meaning in the clue than in the surface reading.
2. Double definition: In this type of clue, the wordplay is simply another definition for another meaning or sense of the word. Sometimes one of the definitions is a pun. Example: "Pisa's tower does it fat-free." (solution). As in American crosswords, a punny definition is often denoted by a question mark (?) at the end of the clue.
3. Anagram: In this type of clue, the wordplay is a scrambled form of the answer, together with a word (called the anagram indicator) which indicates some sense of rearrangement, mix-up, confusion, or change. Sometimes the letters to be rearranged are given literally in the clue, while other times the word to be anagrammed may be one that is defined or suggested by the clue, but anagramming of words defined in the clue is generally limited to easier words. Anagrams are often combined with other clue types; you could have a container clue where one of the two words is anagrammed, or a charade where one of the parts is anagrammed, or a couple parts are anagrammed together.
4. Reversal: In a reversal clue, the wordplay is a word (possibly literally given but usually defined) together with a word that suggests reversal: back, turned around, left, west, up, north (note that some of these only work with across clues or only with down clues). This is often combined with a charade, where either one part of a charade is reversed, or a charaded combination of two or more parts is then, as a whole, reversed.
5. Charade: In a charade clue, the answer is clued as two or more parts, which can be joined to form the answer. For example, "Find the opposite of solid water." (solution) Charades are very often combined with other clue types, especially in cluing some of the parts that make up the clue. A charade clue does not need any special indicator word to indicate the joining of parts, though sometimes one may be present.
6. Container: In container clues, the wordplay consists of clues for two words (sometimes one of them may be given literally, especially if it is a small word), together with an indicator of one word being within, surrounded by, eaten, or otherwise contained in the other word. Thus "gaits" could be clued as "it" inside "gas". Often one or both parts of the container are clued as wordplay of other types.
7. Hidden Word: In these clues, the answer is hidden within other words, as "April" is hidden within "chap riled." An indicator word is normally present, either indicating concealment or containment; these indicators can sometimes be confused with the indicators for container clues. These clues usually stand alone, but occasionally can be combined with other type of clues, such as a reversed hidden word.
8. Homophone: In these clues, another word that sounds the same as the answer is clued, along with an indicator word that suggests sound, speech or hearing.
9. Deletion: In these clues, a word is clued that contains all the letters of the answer in order plus one (or sometimes more than one) extra letter, and some other indicator suggests to remove the extra letter(s). Most often these are beheadments and curtailments, but the deletion clue can also suggest the removal of the middle letter or some specific letter.
10. Letter choice: These are usually not clues on their own, but are parts involved in the other clue types. These clues suggest a specific letter or letters, usually by taking the first, last, or middle letter of some other word, as in "first of March" (M), "last year" (R), and "apple core" (P), or a letter in a specific position in the word, such as "second class" (L). These clues do occasionally appear on their own without other wordplay, as in "Start of each afternoon seems youthful, not difficult." (solution).

The best way to learn to do cryptic crosswords is by studying a lot of clues with their solutions, and then carefully working out how they are constructed. Unfortunately, there are very few opportunities to do that these days. Magazines that regularly run cryptic crosswords, such as Harper's, The Atlantic Monthly, and The New Yorker, do not include instructions with their puzzles; they simply assume knowledge of the rules.

So then. To the abstract principles that /dev/joe described above, I add some extra examples and what I hope are helpful hints.

• Anagrams

  This is by far the commonest kind of cryptic crossword clue, and it should be the first that you look for. As /dev/joe explains above, it will always be signalled by the presence of a word meaning "confused," "mixed up," "messy," "scrambled," "rearranged," or something to that effect. If you see a word that leads you to suspect an anagram, start counting up the letters in nearby groups of words to see if any of them match the number of letters in the answer. For example:

  Crazy-ass ape, key to illegal tavern (9)

  The word "crazy" here suggests that you should be looking for nine letters to rearrange. Punctuation, such as the hyphen in "crazy-ass" and the comma after "ape," are often put into clues to distract you: don't be misled by the clue's strange sentence structure. Rather, notice that "ass ape key" adds up to nine letters, and that they can be rearranged to form the word speakeasy, which is defined as an "illegal tavern." There's your answer.

  Ready for another one?

  Cool boots shuffle! Dance storm! (10)

  This is another sneaky one: reading "shuffle" after a description of boots may make you think of the movement of feet, but the word is actually signalling a mix-up, a shuffling, that is to say, an anagram. What's being shuffled? Well, "Cool boots" doesn't add up to ten letters. But "dance storm" does. Mix up the letters in "dance storm," and you get the name of the coolest boots around.

  Note that in American-style cryptics, the words to be anagrammed will always be included in the clue itself. In British-style cryptics, you will sometimes be expected to anagram a word that is defined but not given directly.

• Charades

  These are probably the second most common type of clue and, as the previous writeup points out, they usually aren't marked by any indicators. Sometimes the words to be joined are given to you directly; sometimes they are defined. Sometimes a clue will combine words with definitions, all of which will be charaded together. For example:

  A housepet, a log, a list (7)

  A housepet is a cat: it's being defined indirectly. But in this particular charade, you are adding it to two words that are simply given to you. Cat + A + log = catalog, which is a list (at least in U.S. spelling; I learned the word as catalogue, but I can't figure out a way to clue it).

  Be careful with the indefinite article. Sometimes "a" and "an" are ignored (as in "a housepet"), while sometimes they are a crucial part of the answer (as in "a log").

  Let's do another one.

  Springtime holiday, we go east to emergency room (6)

  There are lots of little words here which might be used in a charade, such as "we," "go," and "to." Plenty of charades would use these words; this one, however, is getting letters from another source. "Emergency Room" is frequently shortened to E.R. Add an ER to East, and you get Easter, which is a "springtime holiday" (incidentally, the holiday on which I'm writing this -- happy Easter! Here's hoping you don't spend it in the E.R.).

• Combining Anagrams and Charades

  Cryptic crossword enthusiasts know that puzzle constructors will sometimes use common abbreviations to get the letters they need: Right gives you R, University gives you U, West gives you W, Alabama could be AL, Great Britain might give you G-B or U-K, Doctor might give you D-R or M-D (or, used as a verb, it might be the sign of an anagram...), and so on.

  The hunt for necessary letters might be tougher still: "head of lettuce" is L, "artichoke heart" is C, "final tally" is Y, "ten" might give you the Roman numeral X, "the capital of Germany" could be G, "even better" might be E-T-R (the second, fourth, and sixth letters of "better"), and so on, and so on. Any of these might be part of a charade, and all of them can be combined with defined words or anagrams to make a kind of rebus.

  Here is an example to show you how such a combination might work.

  Prime Minister allows harm, drunk on white candy (11)

  "Drunk" here could be a sign that you're expected to anagram -- but what would you be anagramming? None of the combinations of words in this clue add up to 11 letters. The trick is that the anagram is only part of your answer. The "prime" letter of Minister is M. Now make "allows harm" drunk by mixing up its letters. M + allowsharm makes marshmallow, which is a "white candy."

• Container Clues

  The indicator for these clues can be either active ("holds," "conceals," "embraces") or passive ("inside," "hidden," "nestles"). Both the outside and the inside word can be defined or given to you directly, just as is the case with charades.

  They devour urban music for analysis (7)

  Here the word "they" does not need to be changed. "They" "devours" a word that is defined for you: "urban music" is rap. The(rap)y is a synonym for analysis.

• &Lit. Clues

  These are rare, presumably because they are difficult to write. In this type of clue, the wordplay and the straight definition overlap completely; you read the whole clue twice, as it were. The name of the clue type means, "... and take it literally."

  I don't know how to write one of these clues myself, so I will use a classic example from Games Magazine.

  Terribly angered! (7)

  "Terribly" is the indicator for an anagram: mix up the letters in "angered" in a terrible way, and you get enraged. That's the wordplay. Now take the whole clue literally: enraged means "terribly angered." &Lit clues are always marked with an exclamation point, though the presence of an exclamation point does not necessarily mean that your clue is &Lit.

• Double definitions

  Like &lit clues, these are usually very short; normally they only need two or three words.

  Universal brass (7)

  General is both a word meaning "universal" and a word meaning "brass." It is easy to be misled here: most people will think of metal when seeing the word brass, but the clue is aiming for a more specialized meaning -- a military one in this case.

  Puns infect many cryptic crossword clues, and they're frequently found in double-definitions. It is considered good form to mark puns with a question mark. I particularly like this example, which I borrowed from the New Yorker's guide to cryptics:

  Ahead of time, like a nobleman? (5)

  The answer is "early" -- a word that means "ahead of time" and which could, in a silly sense, mean "resembling an earl."

• Homophones

These clues are easy to spot, since they are always marked by a word that refers to speaking or listening ("tell," "announce," "hear," "the audience," etc.).

They say baby birds made beer (6)

"They say" will tip you off to the fact that this is a homophone. So now you have to think of a word for "baby birds" that sounds like a word for "made beer." Brood/Brewed will do the trick. If you're not sure which one belongs in the grid, remember that the answer is always either at the beginning or at the end of the clue: the word you want in this case is "brewed." (Also, "brood" is only five letters and the word you want is six -- though in some cases the word-lengths are the same so you can't depend on them.)

Though all cryptic crossword clues will sometimes use puns, homophone clues are especially conducive to punning.

Makin' leather, talkin' 'bout that stuff in wine (6)

The "talkin' 'bout" is the indicator for the homophone here. "Makin' leather" could be "tannin'" -- a sloppy pronunciation of the word "tanning," which in turn sounds like a chemical that is frequently found in wine (tannin). Though it is usually considered polite to mark puns with a question mark, it is not strictly necessary here, since the unconventional spellings warn you that something punny is happening in this clue.

• Reversals

  These are marked with a word suggesting backwardness; as /dev/joe explains, the geography of the grid itself might be taken into account, so a Down clue with a reversal might use the word "up" or "rise," and an Across clue with a reversal might use the word "west" or "left." It's more common, however, to use generic words like "back," "turned around," and so forth.

  Wrongdoing undone, slower than a gallop (4)

  A legal term for a wrongdoing is tort. If you "undo," which is to say reverse, the tort, you get "trot," which is "slower than a gallop." Since both the original word and its reversed form are always going to be the same number of letters, the clue is required to give you instructions on which of the two words you are expected to enter into the grid. In this case, the grammar of the sentence makes it clear that you are reversing the word for "wrongdoing," meaning that trot, not tort, is what you write down.

  Reversals can be combined with other types of clues, particularly charades. Here is a slightly more complex clue.

  Poisonous snake in commercial is the opposite colour of blood (5)

  "The opposite" is the indicator for your reversal here. Reverse the word for red, the colour of blood, and you get the meaningless DER. But don't despair -- elsewhere in the clue you can find help. A commercial is an ad, and AD + DER gives you the name of a poisonous snake.

• Hidden Words

  The trickiest thing about these is that the indicators for them are often the same as indicators for container clues. As the previous writeup explains, hidden-word clues are rarely combined with other kinds of clues, so once you know that this the sort of clue you're dealing with, they're very easy to solve.

  Brad is hungry, eats root vegetable (6)

  "Eats" could be the sign of a container clue, but in this particular case it's pointing to a hidden word instead. Look carefully into the first three words: bRAD IS Hungry conceals the name of a "root vegetable."

• Deletions

  Usually indicated with a word like "beheading," "endlessly," "heartless," "cut short," or what have you, this is a way to take some letters away from a longer word, and working with whatever remains. Sometimes this will be the only wordplay you get.

  Prune, please, for long-term rental (5)

  Here you are expected to "prune" the word "please" by removing a letter from it. You're left with the word lease, which means "long-term rental."

  Other times, a shortened word will then be anagrammed, used in a charade, containered, or some combination of the three.

  Abridged book on Chinese dynasty is making a loud noise (7)

  "Abridge" a book by removing its final letter: book turns into boo. But you're not done. Now you need a word for a Chinese dynasty: how about the Ming? BOO + MING = a word for "making a loud noise".

• Surprises!

  Crossword constructors are known to be fiendishly clever, and they will often combine clue-types in creative or surprising ways. However, the rules of crossword construction require that the writer always gives you some sort of hint about what he is doing. I've encountered a handful of clues in published sources that struck me as obscure or unfair, or that I couldn't understand even after their logic was explained to me. But usually each clue provides you with all the tools you need to solve it.

I think I'm getting the hang of it. What do I do now?

I find that Games Magazine, along with its sister magazine World of Puzzles, tend to have the easiest cryptics. Unlike the puzzles in Harper's and The Atlantic Monthly, the Games cryptics usually do not have a "trick" to the puzzle as a whole; you just solve the clues and stick 'em into the grid. By contrast, the puzzles the in the other magazines I named often require you to manipulate the answers in some further way before fitting them into the grid. This means that the few answers you do get might not be very helpful for filling in the blanks in your further work, and in particularly cruel puzzles, you might not be given word-lengths to guide you. I love the puzzles in the monthlies -- Harper's especially -- but even now, twenty years after learning to do cryptics, I can only finish about a third of them.

There are few things more satisfying than manipulating letters according to oblique instructions, and then suddenly having a word appear, just like magic.

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Edited to Add:

• WaldemarExkul informed me that The New Yorker no longer runs cryptic crosswords. Alas, indeed.
• tdent explains that the word "regularly" is sometimes an indicator that you are to take alternating letters; the example he gave was Bother if purses checked regularly (4). I don't know if I've ever seen this clue type... or maybe I have and I just couldn't figure out the answers!