Earth-Three is a mirror image of Earth-One, with historical events happening in reverse to what happened there and with a team of super-villains as it's Justice League analogue, called the Crime Syndicate of America.
Earth-Three was created by Gardner Fox and Mike Sekowsky.
Silver Age / Earth-Three
Earth-Three only had one superhero, Alexander Luthor, who attempted to fight back against the CSA. This original version would remain until Crisis on Infinite Earths wiped out the DC multiverse.
Post-52 Weeks / Earth-3
Although the Crime Syndicate appeared in various forms after this, (see the anti-matter universe) Earth-3 properly wouldn't reappear until the multiverse was revealed in 52 Weeks to have been recreated at the end of Infinite Crisis. This version had the Crime Society of America, and was an inverted version of Earth-2. This Earth had more heroes, including heroic versions of the Joker and the Riddler.
The New 52 / Earth-3
Post-Flashpoint, Earth-3 was reintroduced at the end of Trinity War, and was the primarily shown in Forever Evil. It is shown to have a reversed geography compared to Prime Earth, and is said to be where true evil originated. It's signature super-villain team was simply called the Crime Syndicate here, and featured new members such as Atomica and Deathstorm. It had a heroic Joker, and a Alexander Luthor/Shazam hybrid called Mazahs. This version was decimated by the Anti-Monitor.
Infinite Frontier/ Crime Syndicate series
After their destruction and rebirth in the battle against the Darkest Knigth, earth 3 reappered with a new Crime Syndicate and with some changes, stetic and conceptuals. Ultraman resembles most to Antimatter's Ultraman, Owlman is more close to Crisis in two earths version, Superwoman is an alternate version of Donna Troy instead Princess Diana or Lois Lane, and Power Ring, now rebranded Emerald Knight, is member of a corps of conquerors, more unwilling to be as criminal as his peers. Also, Luthor is more heroic and selfless than his previous iteration.
In Convergence, the Silver Age Earth-Three city of Metropolis is trapped under the dome on another planet.
Justice League: Crisis on Two Earths
Earth-Three was shown in the animated movie Justice League: Crisis on Two Earths. This version was shown to have a fairly large Crime Syndicate, although the core of the team is still the same size as usual. It also had at least two heroes, Alexander Luthor and the Jester.