An Enigma machine is a rotor device used to encrypt messages. It was developed by a German engineer and the Wehmacht Enigma, used by German military during WWII was the one solved by Alan Turing.
It's a complicated but all-together basic electro-mechanical sytem. We can imagine it starting with a basic one-to-one encryption; ie. each letter standing for the one before it in the alphabet (A=>B, B=>C). There is a qwerty keyboard for entering code. When a letter is pressed, a rotor advances such that the letter would now represent the one 2 spaces before it in the alphabet (A=>C, B=>D). Except the Enigma has multiple rotors...You can find the details of it's set-up (and find more pictures) on the wiki.
When you're pros with the concept, check out this fun description sparked by a comparison of wristwatches.