The Enigma's trip to British intelligence at Bletchley Park was not the first time any outsider had examined this top-secret machine. In 1928, the Poles had a head start deciphering the code, before Germany invaded their country. (Scroll down halfway for the relevant facts.)
Violating their own security procedures, the Germans sent an Enigma machine to their delegation in Warsaw. When they realized the mistake had occurred, the Warsaw Germans made urgent inquiries about the package they were expecting. This, of course, interested Polish Customs who alerted Polish intelligence. For one weekend, Polish code breakers had an opportunity to thoroughly study an Enigma machine - before it was carefully repackaged for Monday morning pickup.
As a result of this examination, three Polish mathematicians discovered that the Enigma's keyboard was wired in alphabetical order, not keyboard order. With that discovery in hand, they created a decoder machine, called, La Bomba (because it was cylindrical, like a bomb) which helped the Poles trace Nazi naval, air and land movements. By 1939, with a German invasion imminent, the Polish government passed their closely guarded secrets to the British and French.
Already using Polish intelligence, Alan Turing and his team at Bletchley Park were in the process of creating their own decoder - called The Turing Bombe when they received all the data Balme and his team had recovered from U-110. A one-ton machine that measured 7' long, 6'6" high and 2' deep, two hundred Turing decoding machines were built during the war. (Follow this link to a Turing Bombe decoding simulator).
Before they could build a decoder, however, the Bletchley Park team had to figure out how Enigma worked. Follow this link to Nova's site for great drawings and a simple explanation of the Enigma system. Follow this link for a more technical analysis on Enigma and the Turing Bombe.
An eccentric genius who dramatically helped his country during the war, Alan Turing's use of the U-110 information did not give British intelligence an infallible decoding method. The process worked only as long as the Nazis did not change their system. When they DID change their encryption process, three more British sailors risked their lives for additional Enigma data.