Minor Key

Transposing is moving to a different key (C Major to D major).  Changing from Major to Minor and vice versa is called a "change of mode" (C Major changed to C Minor).  C Major changed to E minor is both a change of mode and a transposition.  To get the parallel minor key of the original key, you flatten the third, sixth, and seventh degrees of the key. For example:
E major: E F# G# A B C# D#
E minor: E F# G A B C D 
Here are some examples of songs that have had a change of mode from a major key (written originally) to a minor key.
Hey Jude

Blackbird

Tears in Heaven


Daft Punk's "Get Lucky" switches between A minor and D minor seamlessly.