Is it a Tardis or the Tardis?

The Tardis is an intelligent entity. There are multiple references over the 50 years (since 1963) that the show has been on the air to the sentience of the Tardis, from examples covering its ability to demonstrate free will, to using an avatar to communicate.

The Tardis is not a unique entity in concept. Time Lords used them as they saw fit. As the Tardis that we are introduced to is a TT Type 40 Mark 1, that model designation implies there are other models. In point of fact, The Doctor stole the Tardis from a museum, as it had been decommissioned. So, although it is "a" Tardis, it is the only one we are shown through the series and this is one of the reasons it is considered "the" Tardis.

From Wikipedia

TARDISes are bioships that are grown from a species of coral presumably indigenous to Gallifrey, as stated in "The Impossible Planet", (2006) and it can take years to complete one.

The Tardis appears as a police box all the time because it is damaged. A Tardis normally can and will take different shapes through the power of its chameleon circuit to blend in to the surroundings of the time and place it visits. The Doctor likes it as it is for multiple reasons so this is something he never bothers to repair. This adds to the fact that it is unique, and the reason it is referred to as "the" Tardis.

So both "a" and "the" are correct when referring to it.

This is of course going to beg the ever prevalent question about rationalizing possible conflicts in Canon.

Susan in fact states that she picked out the name Tardis in "Unearthly Child" but as the series has continued, they are all referred to as Tardis. "The Master's Tardis" and "The Rani's Tardis"

So in theory we have three simple potential answers:

1. Susan was acting a bit like a teen-ager and stretching the truth
2. Susan is in fact the Time Lady responsible for the naming of the Tardis
3. Tardis is the actual Gallifreyan word and Susan is responsible for the English Backronym that helps humans understand a LITTLE about the nature of the Tardis.

Or at least 3 would be a great theory if the line didn't indicate the opposite: "Well, I made up the name Tardis from the initials, Time And Relative Dimension In Space." - Susan, "An Unearthly Child" (transcript)

This would throw down my least favorite gauntlet in the realm of "Unobtanium Magickal Translation Systems" or UMTS for short

Let's assume that the ship is in fact called the 'Gryf Mobe Nopre Peen Dprew' which is High Gallifreyan for "Time and Relative Dimensions in Space." And that Time Lords affectionately refer to the ship as a GMNPD. The Tardis translation circuit (a UMTS put to the test in "Fires of Pompei") would have to look thru time for the first Time Lord/Lady to devise an acronysable (is this even a word) translation; so that when a human heard the word GMNPD; the Tardis would translate that so in fact a human hears "TARDIS." Consider who would be speaking in the language of the Time Lords and who would be speaking English and who the Tardis is translating for. This gets confusing enough.

Granted, the High Gallifrean word might in fact be a telepathic word that 'sounds' like "Tardis" which would simplify everything but would be of very high coincidence. And the Doctor doesn't like coincidence.

This entire massive plot hole created by the UMTS is brought to you by the "Klingon Traditional Battle Sword" which the Universal Translator always calls a Batleth. And by America Ninja Warrior's Mount Midoriyama which translates to Mount Green Mount.

It's definitely a TARDIS, and other TARDISes have now appeared or been mentioned in the show.

• At the end of Series 8, the Doctor mentions that Missy (the Master) must have a TARDIS of her own:

  DOCTOR: Dead bodies don't have minds, of course, but she's been upgrading dying minds to a hard drive for a long time. So she upgrades the hardware, and then she updates the software.
  KATE: What do you mean, a long time? How long?
  DOCTOR: Well, she must have a TARDIS somewhere, so as long as she likes. The past, the future ...

  _{-- Death in Heaven (emphasis mine)}

• And at the end of Series 9, the Doctor actually goes back to Gallifrey and steal a TARDIS (again!) The style of the control room is like that of the First Doctor's, only much whiter and shinier and newer. Eventually, Clara and Ashildr zoom off in the new TARDIS while the Doctor goes back to his old phone box, and we get this awesome shot of the two TARDISes whizzing through space together:

  

For the explanation of the apparent discrepancy with Susan's statement way back in the very first episode, see Andrei Freeman's excellent answer.

Yes, and yes. This is more of an English question instead really.

The TARDIS and a TARDIS simply depends on your reference point. It's exactly like I have the car, meaning a specific car, vs I have a car, meaning a nonspecific car of many.

In the new Who, it's simpler as only the Doctor's TARDIS is supposed to exist, making the distinction pointless, but as of series 9, and The Doctor's Wife, and the whole of classic Who, every time lord has a TARDIS. But as the show is about The Doctor, the TARDIS refers to his specific one, 99% of the time.

There are multiple kinds, Battle TARDIS with weaponry to fight Dalek for example. The Doctor's TARDIS is a relic, old, outdated, passenger model, like an 1980 Pinto, rusted and always breaking down.

Keep in mind that TARDIS is not english, as everything we hear is translated for us. The Doctor is not speaking about it in its original Galifreyian language. The TARDIS is translating it on the fly or The Doctor is speaking English.

When Susan said she named it. The "Name" TARDIS could have been a redubbed name for the class of ships. Think "Spruce Goose" and etc that is not "the" name but "a" name