What does it mean when one civilization "denounces" another?

It's not in the manual, but this discussion on the Civ Fanatics forum has some good info. To summarize:

What does it mean when a civilization denounces me?

It always means that they are less friendly towards you. They may go to war with you. Their allies or other "unrelated" players may also denounce you or go to war with you.

What effect does it have when I denounce another civilization?

The same basic effect as when a civ denounces you. Your allies may join with you in denouncing them and might attack them, and your relationship with the denounced civ (and possibly their allies) suffers and they may retaliate.

What kinds of actions can you take that would cause another civilization to denounce you?

War on their allies; war on unallied city-states; taking most of the useful territory on a shared continent; otherwise getting too far ahead of them. Generally anything unfavorable for them, since the AI is stacked against the human player.

In addition to Matthew Read's answer, your standing with non-allied/neutral civs that are already mad at your target will improve. That, in turn, may lead to some unexpected feelers/offers.