For the original Counter-Earth created by the High Evolutionary see High Evolutionary's Counter-Earth.
After the heroes of Earth seemingly died in battle with the powerful foe known as Onslaught, Franklin Richards unbeknown to himself created a pocket universe for the heroes to save them. The Avengers, the Fantastic Four and Hulk resided in this universe for a number of months.
Counter-Earth was created by the team of Jim Lee, Rob Liefeld and Jeph Loeb and first appeared in Heroes Reborn issue ½ (1996), a special that was published by Marvel Comics and Wizard.
Major Story Arcs
Heroes on a different Earth
The counter-earth had a great deal of likenesses to the original universe where the heroes came from, with a few exceptions. Certain historic events where different, and some people lived while others had died. Bucky for instance was a girl in this reality and still lived. At first the heroes had no knowledge of the way they got to this Earth, and thought it to be the Earth where they had always lived upon. The heroes however soon found out that something was wrong. Member of the Fantastic Four Sue Richards started getting feelings of great loss, eventually resulting in visions of a young boy. This was in fact Franklin Richards, a child they did not know they had, still residing on the other Earth. Because of this, the heroes started researching their current position and eventually managed to return to their original home, reunited with their friends and family. The Counter Earth however remained intact and the people living there remained as well.
Counter-Earth Moves : Enter the Thunderbolts
The Earth from this universe was at one moment moved to the 616 universe after the destruction of the pocket universe, where it still resides, undetectable, on the opposite side of the sun as Earth. However, with the heroes gone, the city and possibly that world went into great turmoil. Eventually the superhero team known as the Thunderbolts where transported to this counter earth and took it upon themselves to shape the world into a better one, with them standing as the first line of defense as the planet's heroes. The Thunderbolts eventually also returned to their own Earth.
More recently, Franklin Richards was tricked into restoring fragments of Counter Earth by a super-villain named The Quiet Man to have the "Heroes Reborn" heroes invade the real earth and fight it's heroes.