In the 1987 cartoon series the Trans-Dimensional Portal is a form of teleporter technology that can open dimensional barriers allowing individuals to travel from one dimension to another with relative ease.
The Technodrome is equipped with a large portal in its main control room. This portal can allow individual people to traverse dimensions or even move the Technodrome it's self. However, to open a portal big enough for the entire Technodrome required a great deal of power. It is an established plot element, especially in conjunction with Dimension X, which can only be reached via a particular Trans-Dimensional portal.The warlord Krang had permanently installed such a portal in his super fortress, the Technodrome in order to get the resources that are still left to him after his fall in Dimension X.
Usually the portal was used to travel between Dimension X and Earth, but has been used to travel to Dimension Z, the Dimensional Limbo, the 2003 Turtles Dimension, and to Turtle Prime (The Mirage Turtles dimension).
Once he got to know the technology, Donatello also built on various occasions much more compact versions of the Trans-Dimensional portal, which can range in proportions of about a TV screen size down to a bulky flashlight
Donatello built a small portal that he kept in his lab in the lair. It never worked very well (leading to him turning it into a large-screen TV in one episode), but occasionally would open long enough for them to get to and from Dimension-X.
Krang created a small portal in the Hall of Science building when he and Shredder had to make a temporary base there. This new portal allowed them to finally return to the Technodrome and begin a plan to finally defeat the Turtles.
Donatello and Michelangelo traveled to Dimension-X in the final episode of the 1987 series. The Technodrome had been long abandoned and severely damaged by this time. They used the Technodrome's portal to return home, but it barely had enough power to operate.
The Technodrome was completely redesigned by Utrom Shredder during Turtles Forever. That included the portal room, making it capable of analyzing the two teams of turtles in order to locate Turtle Prime.
During the Turtles Forever special the 1988 Donatello and the 2003 Donatello built a trans-dimensional portal stick from the parts of a flash light. This allowed them to travel to the 1988 Dimension, the 2003 Dimension, and to Turtle Prime.