Closely Related Keys
Composers in the Western Classical Tradition tend to restrict modulations to a small group of closely related keys. The closely related keys for C major are shown below. If a piece moves into one of these keys for an extended period then the closely related keys for that key as a tonic may also come into play. As shown below, it is also common for pieces to move into the parallel tonic minor and then use those closely related keys as well.
One common way of making the modulation from one key to another smoother is to use a chord that is common to both keys as a sort of pivot. In the example below, the A minor triad in the red box is vi in C major and i in A minor. This sort of modulation means that there is never a chord progression that does not make sense in one key or another.