Renunciation is basically giving up, letting go. It can be on different levels. Giving up some things that you find desirable, pleasant, that are not really useful to you, that block your practice. We try to give up things that are hindering our growth in various areas.
Renunciation can be approached in different ways. Sometimes during a retreat there is a bit of "forced" renunciation. The rules are a forced renunciation, you have to give up conversation with others, you have to give up eating whenever you want. You have to give up coffee and all of these sort of things that are unbeneficial, yet you may still like to do them after the retreat, even if some of them aren't so helpful to you.
There is another type of renunciation that occurs when compassion gets strong enough, when we see this is just Dukkha, we let go and we don't really want it anymore. It's not a strain or a stress to us, or a struggle to give up. This is a gentle, wise renunciation.
One type of renunciation can come from determination, and another type of renunciation can come from compassion and wisdom. Both types are needed on the Path, because sometimes it takes a while for our compassion and wisdom to catch up to our determination and energy.
On another level, the bigger level, renunciation is the letting of greed, hatred and ignorance.