The enigma was an rotor cipher machine used by germans in world war II to encrypt their war relevant messages.
In the standard engima you had to choose three out of five rotors to set in your machine giving 5 * 4 * 3 possibilities.
The right and middle rotor shift the other rotors stepwise further. Depending on which character you start there are 26 * 26 possibilities.
All thre rotors have to be set to a specific three letter code to beginn (eg 'A' 'A' 'A') resulting in 26 * 26 * 26 possibilities.
Because you can interchange 10 letters by switching cables at the plugboard there are 26! / (26 - 2n)! · n! · 2^n more possibilities (check http://users.telenet.be/d.rijmenants/en/enigmatech.htm for further information).
There was a set of three reflectors to choose from. Because UKWB was used in most cases this factor is neglected.
All in all there are 1.07 x 10^23 possibilities to set your machine.