Needless to say, as of this writing (8/14/13), the following information is woefully out-of-date (it makes mention of certain aspects of continuity prior to establishment of the New 52 which are no longer considered canon, for instance).
Hal Jordan: The Silver Age Green Lantern, and arguably the greatest of them all. He was off on ceremonial duties for his superiors when his home city - basically the DC Universe equivalent of San Francisco - was completely obliterated by Hank Henshaw (a.k.a. the Cyborg Superman). In his despair, he fell prey to a parasite called Parallax - a sentient entity comprised of the sum total of fear that's ever been felt in the universe - and slew most of the Corps (although some that he felt were insignificant, threat-level wise, he left for dead) and all but one of his superiors, the Guardians. He later redeemed himself via self-sacrifice to re-ignite Earth's dying sun. For a time, he became the Spectre - the embodiment of the full force of God's wrath on Earth - before being brought back to life a few years ago (in Green Lantern: Rebirth - essential reading, that)...
Kyle Rayner: The Green Lantern of Earth in the intervening years after Hal went insane/was joined with the Spectre. Was chosen at random by a dying Guardian to receive the last remaining power ring (Jordan, as the villain Parallax, had absorbed all the others). Became the hero Ion when that symbiont - the green equivalent of Parallax, albeit embodying willpower instead - bonded with him; subsequently resurrected Kilowog (who trains all the rookie corps members) and the deceased Guardians, and began to build up an entirely new Corps of fresh recruits and old retirees. Ion was forcibly removed from Kyle's body by Sinestro, and subsequently assigned by the Guardians to another Green Lantern who is said to potentially be a great hero in the forthcoming prophecy of the Blackest Night. Kyle is now one of the two Green Lanterns tasked with protecting the Guardian's home planet of Oa.
Guy Gardner: Guy was originally the back-up choice to receive Hal's ring if, for some reason, Jordan had been unable to do so. Personality-wise, he's an often angry, sometimes sarcastic, wisecracking windbag - but he is perhaps the Lantern's most tireless and relentless fighter, he tries to do what's right and is absolutely the one Lantern you'd want backing you in a fight. For a time, he used Sinestro's confiscated yellow Qwuardian power ring; then his DNA was re-written by aliens (turning his body into a living weapon), and he was known simply as Warrior. These alterations were later purged from his system by the villain Parallax; he was eventually issued a new green ring from the restored Green Lantern Corps. Now, along with Kyle Rayner, he serves as an honor guard - tasked with protecting Green Lantern headquarters on Oa.
John Stewart: John was perhaps most famously the Green Lantern featured in the Justice League/Justice League Unlimited animated series. Unlike on that show, however, John never had a relationship with Hawkgirl (instead, his wife was the now-deceased Katma Tui: a member of Sinestro's race, and that villain's immediate successor in the Corps following his expulsion from the same). Interestingly enough, John was a paraplegic in the comics when the television series first aired. Due to renewed interest in the character and his then new-found popularity, however, the writers gave him back his mobility. John inadvertently allowed an entire planet, Xanshi, to be destroyed during his watch. The sole survivor among that world's indigenous peoples - the planet's princess who, at that time, was undergoing off-world combat training - subsequently became the villainess Fatality.