Sometimes, when you beat a video game, you get to play it again, with something changed. Typically, you just get to play the same game over only more difficult. However, some second quests' changes are extensive and well-done, and can double the longevity of one's interest in a game.
The game that (AFAIK) started it all, The Legend of Zelda, had a second quest called just that. The Second Quest in Zelda featured changes to just about everything - the world map, dungeon maps (which were completely different), locations of caves and stores, and the items you obtain. It was substantially harder than the first time through, and was that much more rewarding to beat.
Some games have very different ways of getting you to play through a second time. The possibility of earning a different one of a game's multiple endings are always a way to get the player to try again. Metal Gear Solid had an interesting take on this - depending on which ending you saw, which depends on your actions midway through the game, you recieved one of two items which greatly augmented your ability to kick ass in different ways - one gave you invisibility, the other infinite ammunition. You could then begin the game again from your save file at the end, and retain that item from the beginning. Either of them made it substantially easier to beat the game again, so you could replay the pivotal scene in the torture chamber and try to get the other ending. Granted, this doesn't exactly count as a second quest, since everything is the same except that one item, but the experience is very different the second time around, for obvious reasons.
Chrono Trigger and some other console RPGs have featured a simpler approach to getting the player to seek out different endings - the "New Game +" option, when present, generally signifies the ability to start a game with the statistics and/or equipment from a saved game that has beaten the game already. So your characters are as powerful as they'd normally get in a whole play through the game, and you make short work of the parts that you don't feel like replaying. This doesn't actually add much replay value, but still makes replaying the game more attractive - especially when the game should turn out substantially differently, plot-wise.
I've noticed a couple of different but excellent implementations of similar concepts recently.
Castlevania: Circle of the Moon features the ability, once the game is beaten, to start a series of new games with wildly different abilities. The first seems like a reward, as you start with low attack ability, but with every magic ability granted to you at the very beginning. Subsequent games are even more surprising, though - the third trip through grants you great physical strength, but no magic ever, and the fourth quest weakens both your physical attacks and magic - leaving you with extra-powerful subweapons. Yeah, that's right, after you beat Circle of the Moon three times, your reward is trying to beat it by fighting all your enemies with the Dagger, Axe, Holy Water, and Boomerang.
In contrast, Guilty Gear X2 has a series of replay-encouraging quests that have nothing to do with gameplay and everything to do with story. While it still features the standard fighting game main attraction of arcade mode, where you fight through a similar series of other characters and reach a similar ending no matter who you play as, it also features a surprisingly deep story mode, with a different storyline for each of the 20 characters. The fights are linked by voice-acted dialogue, in Japanese with subtitles, that explains what exactly is going on between all these characters and why they keep fighting each other, something that is usually only clear from reading a fighting game's supplementary information. On top of that, each story diverges into one of three paths, depending on the player's actions (or how well they fight particular opponents), making 60 seperate but overlapping stories that flesh out the background of the game more than the snappy quotes at the end of arcade matches ever could.
In conclusion, more games should have second quests in the style of The Legend of Zelda - the lack of one in A Link To The Past was a tremendous disappointment, though for all I know it could have been restored in one of the subsequent installments that I haven't gotten around to - I never beat Ocarina of Time.