Learn how to play all the basic major and minor scales on your instrument of choice. Listen to the difference between a major and a minor scale, and take note of the fact that each major and minor scale are built off of fixed whole and half step intervals. Don't worry too much about the idea of keys at this point.
Look at a handful of pieces of sheet music. Pay attention to the symbols at the far left of the musical staves. A series of italicized pound signs (#) called sharps or italicized lower case Bs (b) called flats indicate the key signature. This tells a musician what scale the song was written using. If none of these symbols appear, then the song is in either C major or A minor, the two keys that have no sharps or flats.
Learn what keys the sharp signs indicate. C major and A minor have no sharps, G major and E minor have one sharp (F), D major and B minor have two sharps (F and C), A major and F# minor have three sharps (F, C and G), E major and C# minor have four sharps (F, C, G and D), B major and G# minor have five sharps (F, C, G, D and A), F# major and D# minor have six sharps (F, C, G, D, A and E) and C# major and A# minor have seven sharps (F, C, G, D, A, E and B).
Learn what keys the flat signs indicate. C major and A minor have no flats, F major and D minor have one flat (B), Bb major and G minor have two flats (B and E), Eb major and C minor have three flats (B, E and A), Ab major and F minor have four flats (B, E, A and D), Db major and Bb minor have five flats (B, E, A, D and G), Gb major and Eb minor have six flats (B, E, A, D, G and C) and Cb major and Ab minor have seven flats (B, E, A, D, G, C and F).
Decide if the song is in a major key or a minor key. Generally, the first note of the song will tell you if it is major or minor. If the key signature has no sharps of flats, and the first note is A, it is pretty safe to assume that it is in A minor instead of C major. In order to be sure, however, you may need to play a portion of the song and listen to the quality of the music. If it sounds sad, it is minor, if it sounds happy, it is major.
Combine your knowledge of scales and keys and attempt to improvise over a piece of music. If you stay within the key (meaning only playing the notes in that particular scale) most of your improvisation will sound good. Play with other experienced musicians in order to help you learn how to improvise in several different keys.