Privileges are the advantages that somebody enjoys by virtue of being part of some group or other, rather than their own merit; to have these sorts of advantages is to be in a position of privilege. As such, privilege can be thought of as the other side of prejudice, although it is a little broader than that. It is also very closely related to the notion of entitlement, although the two words have interestingly different connotations. The whole idea seems to act as a trigger point for a lot of people, putting them straight on the defensive, which is understandable. When our own privilege is pointed out, it can feel like we are being condemned, or our views are being dismissed, on the basis of things we can't help - which does happen sometimes, and probably shouldn't. It is an important concept, though, and it is worth trying to get past gut reactions.
Privilege is insidious, and worthy of careful consideration, largely because people tend to be blind to their own privilege. If you don't have to experience someone else's problems first-hand, it's terribly easy to dismiss them, or assume that they aren't that bad. This makes a lot of discussions about racism, sexism, homophobia, disability, transphobia and class, among other things, a lot more fraught than they otherwise would be, because they are tinged with incredulity. What, you get sexually harassed almost every week? Surely that must be a gross exaggeration. What, people routinely take you less seriously than they take me purely due to some accident of birth? Pshaw, I've usually known people to be very reasonable, I'm sure it's just your imagination.
Of course, people do sometimes exaggerate their problems, and there is a danger that people might put the way they're treated down to prejudice when there is another, more accurate or complete explanation available. But prejudice, discrimination and inequality in general are far bigger problems than many of us are willing to admit, and stereotyping often runs deep even amongst those of us who insist we see people as equal, so these things are likely to explain a lot more than we like to think, especially if we don't witness them first-hand. If you're pretty sure you're not sexist, for example, check out the experiments which show that most people, whether they consciously think that or not, have strong unconscious associations between femininity and weakness, incompetence, and so on. Even more subtly, it has been shown that women themselves are more likely to demonstrate so-called feminine qualities if they are reminded of their gender, for example with a tick-box at the start of a test asking if they are male or female. These are just a couple of examples of currents of unexamined, frequently unconscious prejudice that run beneath the surface of even the most liberal people's understanding of the world.
Privilege is so often invisible not just because it is so easy to be unconscious of one's own prejudices, but also because it manifests itself so many little ways - as well as the big ones that we get to hear about because they are the main focus of any movement for equality. Peggy McIntosh's short but influential piece 'Unpacking the Invisible Knapsack' is a classic examination of this, mainly in the context of white privilege - which she describes as being 'like an invisible weightless knapsack of special provisions, maps, passports, codebooks, visas, clothes, tools and blank checks.' She spends much of the piece unpacking this, with a list of privileges she has registered in her own life. This is important, because it is only by noticing specific examples that the invisibility of privilege starts to dissolve. To take a few such examples, McIntosh's list includes 'I can go shopping alone most of the time, pretty well assured that I will not be followed or harassed' and 'I can swear, or dress in second-hand clothes or not answer letters without having people attribute these choices to the bad morals, the poverty, or the illiteracy of my race.' In this list of male privileges, we find 'If I do the same task as a woman, and if the measurement is at all subjective, chances are people will think I did a better job' and 'On average, I am not interrupted by women as often as women are interrupted by men.' This heterosexual privilege list includes 'Expressing affection in most social situations and not expecting hostile or violent reactions from others', along with thirty or so other examples. I leave it to you to find or compile your own examples of class privilege, able-bodied privilege, neurotypical privilege, cisgender privilege, Christian privilege and so on.
Privilege is one reason why it is terribly dangerous to allow almost all positions of power in this world to be occupied by rich, white males. It isn't that being rich, or white, or male makes you a bad person - that would be a radical misinterpretation of the concept of privilege - but it does make it terribly difficult to understand the problems of those less fortunate than yourself. It is probably also fair to say that most people tend to avoid taking actions and positions which would threaten their own positions of privilege. This is why we see the same Tory MP who had no problem with claiming £749 expenses for a fancy television for his second home making strident, apparently straight-faced condemnations of people on benefits 'playing the system' after he became Britain's Employment Minister. This is why those of us who aren't rich should be horrified - for good, practical reasons, not just on principle - that almost half of the people in the American Congress are millionaires, and why feminists are right to be deeply uncomfortable that it includes only a third as many women as the rate in the general population.
You will probably find, in your own life, that people occasionally accuse you of not really getting their problems. Try not to take it too hard - you probably don't, unless you've been through the same thing, or maybe if you've spent some time listening carefully and thinking about it. That's not an indictment of you or your character - it's hard understanding the crap that other people have to put up with! They probably don't completely understand your problems either, but it's usually best to avoid getting into a competition about who suffers worst or fails to understand the most - it's not really about that, and people tend to feel like you're negating their grievances if your response is to keep emphasising the problems that other people also have.
The main point here is that our own positions of privilege are very rarely obvious to us, and that makes it very easy to be unfair to people who don't share them. If you are interested in 'going from pro-equality in spirit to pro-equality in deed', as this Shrub article puts it, be aware that it is not as easy as we would like it to be. Of course, you may not be interested in investing time and effort in this at all - and that, as they say, is your privilege.
With thanks to TheLady for feedback on an earlier draft, and Aerobe for providing some handy links and ideas to get me started.