Time and Relative Dimensions in Space
The TARDIS is a space craft that can take the shape of anything in the area to disguise itself. The TARDIS was not built, but grown on the Doctor's home world, Gallifrey.
The TARDIS is stuck in the form of a blue box, or police call box. Police boxes were used for emergency calls and to temporarily store criminals. The Doctor says he likes the new look but in reality he couldn't figure out how to fix the chameleon circuit.
The TARDIS has the magically ability to get inside the travelers head and convert any known language into understandable human language, typically English.
The TARDIS is the most badass space craft ever built. It can travel to any point in time and space, including the end of Earth and the end of the Universe. It can, when properly calibrated or during malfunctions, travel to alternate universes and even the space between universes, although that has been known to unravel the fabric of time and space.
The TARDIS is larger than a city on the inside. It is so large the Doctor has gotten lost in it. But on the outside, Oh my my, it is smaller than a lawnmower shed. The TARDIS is another dimension in a blue box.
Without the TARDIS, the Doctor would be just a cheery but sad man with a screwdriver; a sonic screwdriver.