The TARDIS' Chameleon Circuit broke in Doctor Who's first episode, and it's been stuck in the shape of a police box since. When Doctor Who was first being created back in 1963, the writers struggled to figure out what the Doctor's time machine should look like. Eventually they hit upon the idea of a police box, a common sight in London at the time. Of course, by 1969 radios were making police boxes obsolete, and the last box in London was removed in 1981.
The police box is now so strongly associated with Doctor Who that the BBC own the trademark. The broadcaster filed the trademark shortly after the passing of the Trade Marks Act 1994, although they initially faced a counter-claim by the Metropolitan Police; the Patent Office rightly judged that most people associated the boxes with a wandering, enigmatic time traveler rather than with Dixon of Dock Green. Since then, amusingly, the Metropolitan Police have happily dined out on the Doctor Who connection in marketing, so the judgment clearly worked out for everyone concerned. But why is the TARDIS in the shape of a police box in the first place?
In-universe, the TARDIS has what's called a "Chameleon Circuit," which is supposed to allow it to hide in plain sight. The idea is that, when the TARDIS materializes in a given location, its scanners assess the area - perhaps even using its telepathic circuits - and it identifies an unassuming appearance that will mean nobody realizes it's there. Unfortunately, the Doctor's TARDIS is a battered old Type 40, and the Chameleon Circuit broke down when the Doctor was on Earth in 1963. The TARDIS disguised itself as one of London's 700+ police boxes, and it's been stuck in the shape pretty much ever since. The Doctor has attempted to fix the fault on occasion, most notably in Tom Baker's regeneration story "Logopolis," but in truth she's come to rather like this distinctive appearance.
This is the accepted history, but it's entirely possible a retcon is coming. Current Doctor Who showrunner Chris Chibnall seems determined to rewrite the show's lore, as demonstrated with the Timeless Child retcon. Doctor Who season 12 introduced a never-before-seen incarnation of the Doctor, played by Jo Martin, and it's currently unclear how she fits into the Doctor's personal timeline. Her introductory episode, "Fugitive of the Judoon," appeared to imply she was a pre-Hartnell Doctor. Her TARDIS interior bore a close resemblance to Hartnell's, she had Hartnell's habit of referring to the TARDIS as her "ship," and she didn't know what the sonic screwdriver was; the device was invented by Patrick Troughton's Second Doctor, suggesting Martin's predates all known Doctors. Oddly enough, though, her TARDIS was already stuck in the shape of a police box. It's possible all these clues are misdirects, and Jo Martin is the next Doctor; alternatively, Chibnall could have a TARDIS-sized retcon coming in Doctor Who season 13.
From an out-of-universe perspective, the TARDIS' being stuck looking like a police box began as a way of saving money - simply because the show only needed a limited number of TARDIS props. Money is no longer an issue in modern Doctor Who, but by now the shape has become part of tradition, with the TARDIS remaining a constant even as the Doctor's face changes.