There are multiple dimensions during this story. Homeworld is the ultimate realm (our world) but there are many dimensions that have sprung from our own world.
Our own dimension is the mother of all dimensions. From this world all other dimensions spring, like a bunch of leaky faucets. All worlds are linked via our world.
The source dimension. The original comic book stemmed from Turtle Prime and all other TMNT dimensions are linked to the dark gritty world. If something happens to Turtle Prime, it affects all worlds.
This world is straight from the 80's! Four brave little turtles originate from here and have fought many enemies from different dimensions. Cowabunga!
The 2003 verse is home to the ninja turtles of Fox Kids. In the television movie, Turtles Forever, this version of the turtles met their 1980s counterparts. They are brave, strong, willful and loyal... although they often give their Splinter a headache with some of the antics they get up to!
Between space and time is the Fortress where Lord Simultaneous and Renet live. (Picture from internet background site)
A world that exists between dimensions. The Battle Nexus hosts four-yearly competitions between the greatest fighters of the multiverse. Mike03 won the last battle, but with the new turtle tournament, only time will tell if he can keep his winning title of Battle Nexus Champion!
The live action movies and 2007 CG Animated movie are the same universe. They are known as the 2007 turtles, but also starred in the three original ninja turtles movies. They are much older than their other counterparts and, after having fought many battles with villains, Raph07 and Leo07 have achieved Mastery of Ninjutsu. The world is the most realistic of all ninja turtle realms. (except 2014 'Verse)
This surprisingly square realm is the home of the Nickelodeon turtles. Although young, they have proven their bravery and skill time and time again.
The latest Universe is the most realistic version of the TMNT to date. Using photo-realistic technology, these turtles are insanely detailed. The world is closer to Home World than any other world before (apart from the live action movies) and the turtles themselves are large and terrifying. Although a scary-looking bunch, they still have the same heart as other universes.
The live action show features the only female mutant turtle called Venus. Although the suits are bulky and these turtles find it harder to move than other universes, they embody the spirit of ninjutsu.