The TARDIS is a type of ship created by the Time Lords, a now extinct race of timefaring aliens. The last Time Lord is the Doctor, who also owns and operates the last TARDIS. This article refers primarily to the TARDISes owned and operated by the Doctor.

The Doctor's TARDIS is his second. The first was typified by its broken chameleon circuit which made it consistently appear as a 1950s English Police Box. However, since the Doctor has been able to miraculously and relatively inexplicably spawn a new TARDIS, he has been able to mask his ship's appearance. The new TARDIS is relatively the same as the old one, but with some more reliable functions and some less reliable functions.
Ship and Ship Class History
TARDIS is an acronym meaning Time And Relative Dimensions In Space.
Beginning in the Age of Rassilon, an early portion of Time Lord history, TARDIS began to be used extensively. Their exact origin is unknown, and the Time Lords kept the exact mechanics of the ships secret. What is known is that they are not built, but grown. Presumably this means that the Time Lords did not invent TARDISes, but discovered them.
However the Time Lords came to use the TARDISes, they became the most popular craft among them, used by Time Lords such as the Doctor, the Master, the Monk, and the Philosopher. Their incredibly long term of service has made them easily the most extensively used craft in the multiverse.
The first TARDIS used by the Doctor was stolen shortly after his flight from Gallifreyan law. Brand-new, it instantly bonded to his personality and their relationship was more on the level of lovers than Captain and ship - though this is common for Time Lords when first taking command of an adult TARDIS.
The TARDIS underwent various tortuous adventures disabling several of its functions, including its chameleon circuit. However, it proved reliable with over 900 years of service. It's eventual demise was at the hands of several factions in the orbit of Tatooine in a flash of weapons fire. The Doctor's personal injuries crippled the ship's defensive capabilities.
After a tenure without a craft, the Doctor was able to inexplicably spawn one. This new, baby TARDIS has the majority of the same functions as the previous one, but its behavior is erratic at best. Its term of service has been short, and thusfar successful.
Specifications
TARDISes are capable of traversing time and space through a built-in series of wormhole generators.
Its power plant is the Eye of Infinity, the heart of a collapsed star which can produce tremendous amounts of various kinds of energy, including radioactive, psionic, and solar. This makes all of its components compatible with almost every other technology invented.
The outside of a TARDIS can normally be altered to fit any shape as a disguise. The shape does not need to be larger than the ship's interior, which allows for compact travel on board a vessel with the interior design similar to most large cruisers. The smallest size a TARDIS can attain is just a simple door frame. No matter what form the TARDIS takes, it must always have a door.
All other technology onboard is highly complex, but never discussed in detail outside of Time Lord society, even by the Doctor, regardless of the fact that now he alone knows how TARDISes operate. Despite this, the TARDIS has confirmed defensive and possibly offensive capabilities. Its use as a weapon, however, has never been witnessed.
Crew
Currently the only persistent crew of the TARDIS are the Doctor and Steve the emu.