There are three main types of modulation:
Pivot chord: a chord from the original key belongs to a related key and takes on a new function in the new key. For example, a C Major triad would be a I chord in C Major, but a V chord in F Major. So, if used as a pivot chord, it would be treated like a V chord and lead to F. Or, it could be IV in G Major, and be followed by a D Major chord, which would be the new V. The trick in pivot modulations is that more chords are needed to establish the new key and make sure the ear hears that a new note is the tonic. One chord alone is not enough to establish a new key, so the chosen progression in the new key must be long enough and have enough V-I pull to show the key has changed.
A pivot chord could also be a secondary dominance, and the chord following the secondary dominance becomes the new I chord. In the same way, a longer progression needs to be used to show that the new I chord was not just a brief passing tonicization.
The pivot chord modulation is the smoothest since the keys would need to be related somewhat and will have more than one overlapping chord which will weave the two keys together.
Common Tone Modulation: A major chord is followed by a major chord (or a minor chord by a minor chord) that contains one of the tones of the first chord. For example, a C Major chord (C,E,G) could be followed by an E Major chord (E,G#,B), an Eb Major chord (Eb,G,Bb), an Ab Major chord (Ab,C,Eb), or an A Major chord (A,C#,E). (The F and G major chords don't really count since they already belong to the key of C Major.) The new chord will be part of the chord progression for the new key. It does not have to be the I chord, though it often is.
The common tone modulation is pretty chromatic since the new chord itself can bring in more than one note that does not belong to the original key (say C to Ab.) It is often used at the beginning of large sections of songs, but it doesn't have to be. It can be tucked into the middle of another section, but needs some more careful treatment to do it well.
Phrase Modulation: this is basically when, at the end of a phrase, the key is changed abruptly to something new. This is often a modulation up by a half-step or whole-step, and if you go to church you have heard the pianist or organist do this at some point for the last verse of a hymn.
The phrase modulation is rather abrupt. Modulating up a half-step or whole-step brings in a whole set of new notes. It can also be cliche.