The Eyes of the Overworld is a unique NetHack quest artifact. It normally belongs to Monks (who must retrieve it from one of the most difficult adversaries, Master Kaen), but can be wished for by any non-Monk, Neutral character. Beware that non-Monks will be blasted by its power (4d10 damage) when received by wishing, when picked up, when put on, and when invoked. This artifact (along with The Eye of the Aethiopica and Platinum Yendorian Express Card) is why most players worth their salt play Neutral. Neutral characters get all the good quest artifacts.
The Eyes' base item type is a pair of lenses. This means, while wearing them, you cannot also wear the most massively useful thing a hack can have, a towel, or blindfold (either of which is naturally very useful when you have telepathy). On the other hand, this gem will protect you from being blinded[1] from exploding yellow lights, wands of lightning, potions of blindness, cream pies, spat venom, magic traps (the monsters that were just created might still pose a problem), etc. The Eyes being lenses means you can read spellbooks in two-thirds the time (and with a small intelligence bonus) and can search more successfully. Fringe benefits perhaps, but worth noting.
While wearing The Eyes you are granted 'astral vision' (every geek's dream, x-ray vision) which is a sort of permanent (though less far-seeing) clairvoyance. The squares around you are illuminated as would be lit by a lamp. The difference is astral vision penetrates walls, doors, solid rock, and any other obstruction. This is highly useful in trying to find secret doors or trying to navigate the maze-like Gehennom without magic mapping. In the following contrived screenshot, the : squares are illuminated by the Eyes and clairvoyance, while the . squares are illuminated only by the latter. Of course, you are the @.
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Merely carrying The Eyes (as opposed to wearing them) confers magic resistance. While a wonderful trait to have, you should not rely on it as your only source of MR in the endgame as a Monk. The Wizard of Yendor will steal this treasure (and The Amulet of Yendor if you have retrieved it) and then use his dreadful touch of death on you. This will probably happen to every player at least twice.
#invoking this artifact is the same as zapping (or quaffing) a wand (or uncursed potion) of enlightenment. But, as with every quest artifact, you can only invoke every hundred turns or so. With any luck this shouldn't pose a problem for this particular artifact, because the only time enlightenment is urgent is to find out if you can pray or not, and your prayer timeout is usually far longer than the #invoke timeout (meaning if you've prayed since last invoking the Eyes and you cannot invoke again, you won't be able to pray again). Invoking too soon will add 3d10 more turns to the invoke timeout. Invoking is useful for identifying rings and amulets. Put on an unidentified accessory, #invoke, and look for unusual stats like slower digestion, a +2 damage bonus, or your life will be saved.
Finally, wearing this artifact protects you from Archons' stunning gaze. Archons are absolutely nothing to scoff at, so any advantage helps. Being blind also protects you, but it takes some talent to be genuinely blind while wearing the Eyes.
The Eyes of the Overworld is also the title of a 1966 novel by Jack Vance. This is indeed the source of the item's name, according to The Annotated NetHack File. DejaMorgana and liveforever highly recommend this novel's series to any roleplaying game player.
[1] - The Eyes do not actually prevent you from being blinded. They nullify all of the effects of blindness. If you are blind and put on the Eyes, you will be able to see. If you are blind (it will not show up on the status line) and you remove the Eyes, you will then suffer all the effects of blindness. For most intents and purposes though, saying the Eyes prevent blindness suffices.
Information current as of 3.4.3. Who knows what the DevTeam will come up with next?