The Turing machine is a hypothetical machine (even though people afterwards built implementations in the real world). It was used by Turing to prove that there are "well-formed" problems in computer science that are not solvable. He did this based on work by Gödel and Kantor. This work of Turing predates the development of the electronic computers. It can be best seen in the context of the so-called Grundlagenkrise in Mathematics that addresses the question whether we can prove that Mathematics is true and the resulting Hilbert program that asks to find an algorithm that can decide the truth of an arbitrary statement in first-order predicate calculus applied to the integers. Gödel in 1931 proved that no such effective procedure can exist (incompleteness theorem).

The crux is in the definition of an algorithm or effective procedure. Turing came up with a machine, the Turing machine, that simulates a mathematician executing an algorithm. The mathematician has a limited memory for rules and a potentially infinite tape to do his/her calculations. The point is that everything that a Turing machine can do can certainly be done by a mathematician given unlimited time and paper. This is pretty obvious given the definition of the Turing machine. The other direction is actually more interesting and not so obvious, namely, that everything a mathematician can do can be done by a Turing machine. Since there is no exact definition of the capabilities of an abstract mathematician, this part cannot be proven exactly, but it can be made plausible. Turing was able to show that a Turing machine can do arithmetic manipulations on integers. Thus, Turing can now convincingly define a yes-or-no question on numbers as calculable if and only if a Turing machine can come to a decision (answer) on this question.

Turing then recognizes that a Turing machine can be emulated by another Turing machine. Indeed, there exist a universal Turing machine that can emulate any other Turing machine. The universal Turing machine needs a program as input when it simulates a machine. This input is simply a pattern of zeroes and ones on the tape.

The next step is using Kantor's diagonalization technique to prove that it is impossible to have an algorithm (i.e. a Turing machine) that determines whether a Turing machine on given input will ever stop.

Turing's clarification of computability and the unsolvability of the halting problem is a great achievement. It predates modern, electronic computers. His proofs have been worked over and the Turing machine defined in lectures and Computer Science books is really a Post Turing machine after the logician Post. It turns out that Turing's approach and several other approaches to computability such as Church's lambda calculus and general recursive functions by Gödel and Herbrand are all equivalent. This result makes it plausible to Mathematicians that these equivalent approaches capture the philosophical notion of computability. That this is indeed so is called the Church Thesis or Church Turing thesis.

Turing's work for the British war effort came after these monumental discoveries and has been the subject of many books and even a maybe not so accurate big-screen movie. Turing is also known for the Turing test, which comes out of a discussion of what it means to think, and in particular, whether it makes sense to say that a machine thinks.

The details of the story that I have outlined can be found in books on the history of modern Mathematical Logic or Theoretical Computer Science. Turing machines are still taught at the undergraduate level in various, but not all computer science programs. As computer science attracts the nerdy and eccentric, there are several physical implementations of a Turing machine, which combine often an impressive engineering and machining with a senseless objectives, since of course a Turing machine in the physical world is not a very good computer. It is however "Turing" complete, meaning that any computer today in existence with the possible exception of quantum computers can be emulated on it. Mainly for pedagogical purposes, a number of Turing machine simulators have been built (it is not very difficult) and made available through a web interface.

As /u/tbobsj details, a "Turing machine" is a hypothetical concept developed by Turing before the war; the Bombe and Colossus were two (very different) types of machine constructed at Bletchley Park during the war to break Enigma and 'Fish' codes respectively.

Colossus was the first programmable electronic computer, built in 1943 to attack the Lorenz teletype cipher known as 'Tunny' to the British; it was a specialist device rather than a Turing-complete general purpose computer. Turing had little to do with its design and construction, though it likely inspired his post-war work on the Automatic Computing Engine at the National Physical Laboratory.

Turing was responsible for the British Bombe. Polish codebreakers had, before the war, cracked Enigma, and constructed an electro-mechanical machine known as a 'Bomb' (Bomba/Bombe) that, broadly, replicated a number of Enigma machines to attack a particular weakness of German procedure (repeating a three letter indicator at the start of messages). Shortly before the war the Poles shared their knowledge with the French and British intelligence services, who had made no progress against Enigma, and Turing developed the idea of a mechanical device. Similar to the Polish machine Turing's Bombe replicated a number of Enigma machines running in parallel, but rather than working on the indicator (repeated indicators were dropped in 1940) Turing's used known plaintext (a 'crib') to attack encrypted messages.

Bletchley Park/The National Museum of Computing have further information:
Bombe History from TNMOC
The Bombe Breakthrough at Bletchley Park

The late Tony Sale's Codes and Ciphers website also has more, including contemporary documentation.

It's also very important to ignore 2014's The Imitation Game; though it works dramatically as a film, as history it's utter drivel and gives the impression that Turing himself hand-built the first Bombe without any real idea how it might work (until a chance conversation), and that he called it Christopher (he didn't; the first machine was codenamed Victory), amongst numerous other inaccuracies.