The Turtle Van has a variety of origin stories, most of which are very similar.
1987 Animated Series
The Teenage Mutant Ninja Turtles confiscate Baxter Stockman's van after defeating him, a short time after that Donatello makes some modifications to the van and christens it "the Turtle Van".
2003 Animated Series
After a run in with the Purple Dragons, the turtles end up stealing the dragon's already stolen van. After a while Donatello revamps the van and renames it 'the Battle Shell'.
2012 Animated Series
The newest incarnation of the Turtle Van isn't a van at all, rather a subway car which formerly served as Leatherhead's hide out, it was transformed into a vehicle capable of traveling on and off subway tracks and renamed 'Shellraiser'.
Every incarnation of the Turtle Van has had a large variety of high-tech gadgets and weaponry to aid the Ninja Turtles in their crime fighting adventures. Just a few of the vans many features: Missiles, GPS, oil slicks, water cannons and armor plating.