A multiverse or multi-verse is an infinite number of universes in which all possible events occur.
In the TMHT Adventures story "Turtles of Terror", published in late-January 1991, Krang and Shredder used the Technodrome portal discovering an alternate Earth, whose counterparts of the turtles were criminals. They were sent to the good turtles' Earth to defeat them, but were defeated by the good guys-turtles at the end, and taken back by Krang and Shredder.
Some TMNT stuff was always meant for grownups.
The Multiverse is first hinted for the Mirage turtles at when Leonardo visits the Battle Nexus in volume 4. Discussion about meeting one's other-world counterpart is present, and it is implied that Oroku Yoshi is a counterpart of the mainstream universe's Oroku Saki.
The Multiverse is then shown more explicitly when Galactose rips a dimensional hole, and the Turtles witness themselves in other incarnations, including the Archie and 2007 film universes. Furthermore, Cudley the Cowlick, who appears in a few Mirage issues dealing with the Multiverse, is actually the Archie version, crossed over to the Mirage Universe, instead of a Mirage version of Cudley.
In 2003-2009 series the multiverse consists of different alternate universes and realities were shown to exist as apart of the show's lore. In the show we see in The Big Brawl, Part 1, that Multiverse often converges with each other and allow others to meet each other as seen with Battle Nexus is said to be the center or "nexus" of dimensions connecting many different worlds as shown including Earth which is referred to as Third Earth. Other worlds shown are when Ultimate Drako used the Time Scepter to send the Turtles across different realities, such as the Super Turtles' reality in Reality Check, the world and characters of Peter Laird and Jim Lawson's highly regarded "Planet Racers" series in Across the Universe, a bad future version of the Turtles' reality is seen in Same As It Never Was, and Usagi's feudal world in The Real World, Part 1.
In the Turtles Forever animated TV-movie, the multiverse becomes an important plot point. The Utrom Shredder discovered its existence through the Technodrome's computer, as well as many TMNT alternate universes, including the 2003 series the 1987 series. Wanting to destroy all Ninja Turtles the Shredder seeks the Teenage Mutant Ninja Turtles comic turtles from Mirage. This universe was known as "Turtle Prime", the source where the other realities branch off from, and its destruction (specifically its prototype Turtles) would end all life of the multiverse. A scene in the movie also implies that other Turtles media is part of this same multiverse, including the Live Action films, the 2007 film and Mutant Turtles: Chōjin Densetsu Hen.
The multiverse was confirmed to be part of the Multiverse in the episode The Manhattan Project, Part 1, when the 2012 Turtles observe their counterparts in the 1987 show's universe. They later actually meet these counterparts, and the two Donatellos meet their Mirage versions.