The story
The bombe grew from a Polish machine. In 1938 the Cipher Bureau in Warsaw built the bomba, six Enigma replicas driven by a motor, which hunted for the daily key using a habit of German operators at the time: they sent each message key twice. When the Germans dropped that habit, the Polish method stopped working, and in July 1939 the Poles passed what they had to the British and French.
At Bletchley Park, Alan Turing designed a new machine that needed no such habit: only a crib, a stretch of probable plaintext. The British Tabulating Machine Company at Letchworth built it, under its engineer Harold “Doc” Keen. The first, named Victory, arrived in March 1940. Gordon Welchman then saw that the plugboard swaps letters in pairs and added the diagonal board, which wires that fact into the machine and cut the number of false stops dramatically; bombes with it were working from the late summer of 1940.
Menus, stops and Wrens
Codebreakers in Hut 6 and Hut 8 chose cribs and drew menus; the bombes were run mostly by members of the Women's Royal Naval Service, at Bletchley and in outstations nearby, who set up the drums and cables and reported stops. A stop went to a checking machine. If a few groups came out as German, the key was nearly found, and the rest of the plugs were worked out by hand. By the end of the war Britain had about two hundred bombes, and the United States had built more of its own design.
The rebuild
The wartime bombes were dismantled after the war. A team of volunteers led by John Harper rebuilt one from surviving drawings; it first ran in 2006, was officially switched on in 2007, and can be seen working at The National Museum of Computing on the Bletchley Park estate; the drum colours on this platform follow it.
How it was broken
The bombe is the breaking, so this is how it thought. Suppose the plugboard sends the test letter to some letter X. Follow the menu: at each link the scrambler at that position tells you where the plug of the next letter must go. Around a loop you come back to the test letter; if you arrive at something other than X, the guess was wrong. Electrically, the bombe tried all 26 guesses at once: current spread through the wiring, and if every guess lit up, the position was impossible and the drums moved on. When it did not, the machine stopped.
Nothing in this needs the plugboard to be known in advance, which is why a dozen letters of guessed German could bring down a key with more than a hundred million million million settings.