This is a little long, but please give it a read through since I spent some time covering the bases for most everything you'll need to know in a quick form.
Ok, what you probably know is that the GL Corps is sort of an intergalactic police force. From the central power battery on Oa (a planet at the exact center of the universe) the Green Lantern rings are powered and this allows their rings to do virtually anything their holders can imagine, which is chiefly to make shapes (they can also do nifty other things like translate, scan things, etc.). The rings major weakness is to anything which is colored yellow (which was kind of silly when originally thought of back in the day, but it was a way to limit this otherwise godly powerful ring. GL:R provides an explanation for it). In order to make the rings work, it is essential that the holder exert their willpower, and also be free from fear. Whilst Batman and Green Arrow and some others have occasionally gotten their hands on a ring and used it, it's extremely difficult and takes everything out of them, so not just anybody can use one. The rings also need to be charged every day in a battery which connects to the main battery on Oa. The GLs must say the GL oath when they do this
In brightest day, in blackest night,
No evil shall escape my sight
Let those who worship evil's might,
Beware my power...Green Lantern's light!
Catchy, ain't it?
The Green Lantern Corps is overseen by the Guardians of the Universe, small blue creatures whose homeworld is Oa. They divide space into 3,600 sectors (Earth is in sector 2814), with each sector having one GL watching over it. Upon the death of a GL, the ring automatically searches for a new suitable wielder of the ring within the sector, the nearer the better. Should a GL ever be unable to fulfill his duties, there is a back-up GL also selected for each sector (which is how Guy Gardner and Jon Stewert became Lanterns originally) so there's approximately 7,200 Lanterns. Over the years, we've been introduced to lots of different alien GLs, the most important of which is a guy named Kilowog (picture). There's also another pretty important one named Mogo, who's a whole living planet (picture).
(One quick note is that the Golden Age Green Lantern, Alan Scott, also has a ring and a power battery and the exact same powers (except it's weak to wood instead of anything yellow), but he's not a member of the Green Lantern Corps. His daughter, Jade, also naturally possesses GL powers without being a member of the Corps. It has to do with the fact that DC comics was originally divided into several universes which were merged in 1986's Crisis on Infinite Earths, and the GL Corps were part of Earth1 while Alan Scott was from Earth2. I haven't read much Alan Scott, but I think his powers are magically based or something. )
Now to some of the history, which admittedly I'm a little shaky since I haven't read it all first hand:
Hal Jordan, a test pilot for experimental aircraft at Ferris Aircraft near Coast City, was the original Earth Green Lantern, getting the ring after a GL named Abin Sur crash landed on earth and died. Hal always had a crush on Carol Ferris, the daughter of the founder the company.
Hal Jordan's original trainer was a GL named Sinestro, a tall headed purple alien who was sometimes considered the greatest Green Lantern, because his sector was completely free of disorder. This, however, was because he lorded over his sector like a tyrant, and eventually he was stripped of his GL ring and banished to the anti-matter universe. On a world named Qward, he developed a ring which functions exactly like a GL ring except in yellow, and he proceeded to be the standard Green Lanern arch-enemy for the next few decades of comics.
In the 1970s, Green Lantern formed a fast friendship with Green Arrow. The two characters shared a single comic, and traveled across America in a truck. During this extended arc, their stories dealt with drugs, racism, environmentalism, and other topics not usually touch by comics at the time. It's considered a classic era of comics, and a turning point in comics being seen as relevent and not just entertainment.
Quick summary of Crisis on Infinite Earths (1986): So like I mentioned, DC used to be comprised of several different universes. Some characters were from Earth1, some from Earth2 (several characters had different versions on both) along with Earth 3, 4, 5, Earth X, etc. etc. This sucked because if you wanted Blue Beetle to team up with Batman or something, you first had to have a character cross dimensions. And on one world, Superman and Batman were pretty much like you know them, and on another Supes was older and married to Lois, and Batman was married to Selina Kyle and they had a daughter who was the Huntress. It was decided to mash all the worlds together in CoIE. A character named the Anti-Moniter tried to destroy the multiverse, but the good-guy Moniter tried to stop him and assembled all the heroes. Eventually, everything was left as one world with all the characters together, and no one remember that there used to be lots of worlds. In fact, the way DC pretty much worked was "anything that happened before 1986 didn't happen unless we specifically reference to say that it did". However, at the end of CoIE, the older, married Golden Age Superman and Lois, along with Superboy Prime and Alexander Luthor (a younger good guy version) walked off into a pocket paradise dimension in order to give them a happy ending.
Quick summary of Infinite Crisis(2005): Watching from their pocket dimension, Alexander Luthor and Superboy Prime decided the world which had come out of CoIE was actually a bad world after watching certain heroes do lots of messed up things (not the least of which were the events of Identity Crisis which you read). The two orchestrated a series of events which split the universe into multiple worlds again. Superboy Prime ended up going completely villainous, killing several members of the Teen Titans and Green Lantern Corps along with Golden Age Superman (GA Lois died of old Age, and Alexander Luthor was killed by Joker). At the end, Superboy Prime is seen being kept prisoner by the GLC.
(Due to Lawsuit, DC no longer owns the name "Superboy", and thus Superboy Prime is now called Superman Prime)
Back to the history of Green Lantern: In the early 90s, as you may remembered, Superman died fighting Doomsday, and four replacement Superman's showed up claiming the legacy. One of these was the Cyborg Superman. While playing at being a hero and the true superman brought back from the dead, CSM was actually a villain named Hank Henshaw, and working with the villain Mongul. Their first act of terrorism was the complete eradication of Coast City, Hal Jordan's home city.
(I'm spending a bit more time on this than I planned and I'm short on time, so I'm going to finish this by copy/pasting wikipedia)
In the controversial 1994 Emerald Twilight storyline in Green Lantern vol. 3, #48-50, the villainous alien Mongul comes to Earth in a plot to take advantage of the death of Superman. Jordan defeats Mongul, but not before Coast City (Jordan's former home) is destroyed and all of its inhabitants murdered. He tries to use his ring to recreate the city, but the Guardians condemned this use of the ring for personal gain and demand that Jordan come to Oa for trial. Angered at what he saw as the Guardians' ungrateful and callous behavior, Jordan seemingly goes insane and attacks Oa to seize the full power of the Central Power Battery. The Green Lantern Corps attempt to defend Oa, but the enraged Jordan overwhelms them, crippling his fellow Lanterns (even cutting off the hand of Boodikka and reducing Kilowog to bone and ashes) and the Guardians. He then renounces the Central Power Battery to his life as Green Lantern, adopting the name Parallax.
As Parallax, he initiates the Zero Hour: Crisis in Time, attempting to rewrite history to his own liking, but he is eventually defeated by a gathering of heroes. Jordan is replaced by Kyle Rayner as the Green Lantern of Earth when Rayner comes into possession of the last power ring, created from the shattered remains of Jordan's. During the same storyline, Alan Scott gave up his ring upon the death of members of the original Justice Society of America, and this ring is later crushed by Parallax. Alan Scott soon renounces his "Green Lantern" identity and begins to use the codename "Sentinel". This leaves Kyle Rayner as the sole bearer of the mantle of "Green Lantern."
In the 1996 Final Night miniseries and crossover storyline, Jordan returns to his heroic roots, sacrificing his life to reignite the Sun (which had been extinguished by the Sun-Eater). Many super-heroes, including Superman, view this sacrifice as Jordan's redemption, one final heroic act. Batman is unconvinced, saying that one act couldn't make up for the evil that Hal had committed.
In the 1999 miniseries Day of Judgment, Jordan becomes the newest incarnation of the Spectre. [1] Soon after assuming this mantle, Jordan chose to bend his mission from a spirit of vengeance to one of redemption, also making other appearances through some of DC's other storylines, such as advising Superman during the Emperor Joker storyline (Where the Joker stole the reality-warping power of Mister Mxyzptlk) and erasing all public knowledge of Wally West's identity as the Flash after his terrible first battle with Zoom. A new Spectre series based on this premise, however, lasted only 27 issues before cancellation due both to poor sales and continued calls amongst comics fandom to return the character to his sci-fi roots as Green Lantern.
(Quick notes that Kilowog was returned to life at some point, Jon Stewart's back was broken, and Guy Gardner got... well, cybernetic implants isn't what it was, but it's easier than explaining what the actual story was)
