There are basically two types of 'modulation' (key change).

One, as in your 'Molly Malone' example, is a 'passing modulation' where the new key is suggested, then the original key is restored. It is still technically a modulation, as it happens at the end of a line (cadence point).

The other is a 'true modulation', where the tune literally changes key.

It's not usual for a tune as short as a song verse to completely change key, but it's common for the 'B' music to be in a different key from the 'A' (and possibly the 'C' if there is one).

A good example of this is the 'Skye Boat Song' - the chorus ('A' music) is in G, and the verse ('B' music) is in E minor. This song is a little unusual, because the chorus is more usually the 'B' with the verse being the 'A'...

In this example, G is the major key, and E minor is the 'relative minor'. They share the same key signature, but it is still a modulation as G and E minor are not the same key.

This is also a good example of two ways to modulate. From the Chorus to the verse it is a sudden change, but from the verse back to the chorus, the IV and V chords are used to 'suggest' the key change before it happens.

I'm sorry if this gets a bit too technical...

If you need more clarification, tell me which bits you don't follow, or ask more questions! Examples are always the best way!