So, others may be more qualified to do a proper write up on the T-bird than me but I'll give you my take on it anyways.
In the early 1960's, Fender was a leader in the market and Gibson wanted something premium and modern to compete. Through a chance meeting, a Gibson executive wound up inviting retired automotive and custom coach designer Raymond Dietrich, known for his work with Lincoln, Chrysler, and Checker Taxi, to come and design a guitar. What they wound up with were the "Reverse Body" Firebird guitars and Thunderbird basses released in 1963. There was a Firebird I, III, and V, and a Thunderbird II and IV. Both instruments in their original form used a laminated through-neck construction similar to a Rickenbacker, with a pair of wings glued to the sides of the body, and a slightly raised center section. The Thunderbird II had a single humbucking pickup, volume, and tone pot, while the more premium IV model got two humbuckers, two volume pots, and a master tone. As far as I know all of the Gibson pickups used rails rather than individual poles.
Later on, after a lawsuit brought by Fender, Gibson changed the Thunderbird (and Firebird) models to what's known as the "Non-Reverse Body", which I would argue more closely resembles Fender's "offset" body styles like the Jaguar and Jazzmaster. These instruments offer similar specs to the earlier iteration, but using set-neck construction like a Les Paul or SG. The Non-Reverse birds also have a completely flat front to the body with no raised center section. These were produced from 1966 to 1969.
The Reverse Body Thunderbird IV came back in 1976 as the Bicentennial Thunderbird, and ran until 1979, and I'd say all modern T-birds are arguably reissues and re-imaginings of the Bird.
Famous Thunderbird players include names like Justin Chancellor (Tool), Les Claypool (Primus), Kim Gordon (Sonic Youth), Glenn Cornick (Jethro Tull), Krist Novoselic (Nirvana), Peter Hook (New Order), Kim Deal (Pixies), Adam Clayton (U2), Pete Way (UFO), John Entwistle (The Who), just to name a few.
Today, Thunderbirds can be found from a variety of US and foreign manufacturers, each with their own unique character. Not all use through-neck construction, some are even bolt-ons.
A fairly comprehensive list of manufacturers and models can be found
here.
Departing from the history, I find them to be a fairly versatile bass. They tend to be favored by rock musicians, and play well with a pick and fingerstyle. You can get a fairly diverse range of tones out of them, though they're mostly renowned for a sort of punchy bottom-heavy tone with excellent sustain in many circles. The scale length tends to be 34", with 20 frets, four strings, though short scales and extended range examples have been produced.
The bridge is located a few inches in fron the bottom of the body, and the headstocks can be on the large side, so what results is a standard scale length bass that feels like it;s even longer. If you're used to a P-bass, a Thunderbird is probably going to feel like a much bigger instrument in the hands, which can either be a drawback, or it might be a reason to be drawn to the instrument. They tend to be comfortable to play sitting down as well as standing up. Rumors of neck-dive are slightly eggagerated and the issue that is there is easily rectified with proper strap button placement( which the factory may not actually have done for you sadly) and strap selection.
There's kinda a love it or hate it thing going on with the Thunderbird (and apparently the Firebird too), making it kind of a polarizing name in the bass world.
I like them enough to have three, though. If you're considering getting in to one however, as always I'd suggest going to a few stores and playing a few of them before jumping in headfirst.