I'll try to explain them to the best of my ability.
You are right about Atrocitus being a Red Lantern, though he's not just A Red Lantern, he's the leader of the Red Lanterns. He's an anti-hero whose methods happen to bring him up against the Green Lanterns regularly (His entire sector was wiped out by something that was essentially the fault of the Green Lantern's bosses, the Guardians of the Universe, so he wants to kill all of them).
Cheetah is a Wonder Woman villain, her arch-nemesis (When it's not Ares) to be exact. there's been more than one but the one in the game is Barbara Minerva, who, IIRC was a treasure hunter who went through a blood magic ritual that transformed her into a cheetah woman.
Doctor Fate is a Wizard, an agent for a powerful group called the Lords Of Order. The helmet, cloak, and amulet are all magical artifacts he uses to help fight crime, the helmet containing the spirit of Nabu, the Lord Of Order that created it, though while Fate does use the artifacts, how much free will he has seems to vary from writer to writer. Sometimes he's in full control, sometimes he's like a puppet, etc.
Firestorm is what happens when two people get caught in a nuclear accident, fusing them together to become one being just called "Firestorm". Firestorm's main ability is to alter any inorganic material, but what's holding them back from essentially being godlike is that, at least with the case of the first pair, Ronnie Raymond (A high school Jock) and Martin Stein (A brilliant scientist), is that Ronnie Raymond is, well, an idiot. Though the pair in-game replaces Raymond with the more recent Jason Rusch, who's considerably smarter.
Grodd is one of Flash's villains, a sentient gorilla with mind powers, he's from a city of sentient gorillas called Gorilla City, which (obviously) is a city of sentient Gorillas. Grodd hates humans with a fiery passion, and often attempts to wipe them out, but The Flash keeps getting in the way of that.
Black Adam was originally Captain Marvel/Shazam's Arch-nemesis, though modern depictions have him as a more a Lawful Neutral/Lawful Evil anti-villain. anyways, Adam was chosen by the Wizard Shazam to be his successor, and depending on the version, The power either got to Adam's head (Old version) and the Wizard banished him, or The Wizard perceived Adam forcefully overthrowing the government of his land and installing himself as ruler as Adam being corrupted by a demon. Either way, Adam ends up imprisoned/banished until the modern Era, where he fights Captain Marvel, however, in the Modern Era, Black Adam goes on to join several teams, one of which being the Justice Society.
Jaime Reyes is the third person to take up the mantle of Blue Beetle. It all started when Jaime found the scarab that the first Blue Beetle possessed, which was thought to be a magic artifact, but was really just a piece of technology created by a race called The Reach, who sought to have it possess Jaime, but it gradually developed its own will and independence with Jaime's help.
Swamp Thing is a complicated one so I very well may get details wrong: He was originally a scientist named Alec Holland, who had developed a restorative formula for plant life, but was attacked and killed by someone who wanted the formula with a bomb, Holland ran into the swamp, covered in burning chemicals from the fire, and falls into the swamp, mutating into a strange, monstrous creature that comes to be known as the Swamp Thing. Though it was later retconned that Swamp Thing was a plant elemental already that had somehow absorbed Holland's memories and personality upon his death.
Grid, from what I understand, is the Crime Syndicate (Earth-3's evil Justice League)'s Cyborg, a sentient computer virus that wants to be human more than anything.
Vixen is a woman from Africa who inherited a magic totem that gives the wearer powers of various animals (Cheetah's speed, Hawk's flight, Ant's strength, etc), that was passed down from generation to generation in her family line.
Reverse-Flash is the top contender for Flash's arch-rival: The Reverse-Flash is the man responsible for the murder of Barry's mother, the catalyst that ultimately lead for Barry to become the hero that he is nowadays, attempted to murder his wife, and ultimately forced Flash to kill him to save his new Fiance. But upon being resurrected, he got worse: Reverse-Flash murdered his brother (Who he thought was holding him back), his parents (Due to them not supporting his Speed Force obsession), a professional rival, and every woman a girl he was interested in ever dated. During Flashpoint (The crossover that caused the New 52 which is evil enough IMO) he alters the time stream further, to make Barry miserable.
Power Girl, from what I know, is an alternate-universe Supergirl. From what I understand, she's a businesswoman, and the leader of her world's Justice Society