I've had two situations so far:
one, an AI masses troops at the border (Russia in my Spain game, India in my Sumeria game). And then they... don't do anything. Its obvious they want to declare war, but they seem to uncertain if they'll win, so they just mill about.
two, an AI just randomly declares out of nowhere. This happened in my Spain game, from Sumeria. I was... a lot ahead of them, and they didn't have a lot of troops nearby. It went poorly for them.
On the other hand, AI vs AI, they were a bit crazy. China fed on the Aztecs quite a bit (took 3 or 4 cities), and France took at least two Japanese cities. Sumeria also went to war with the French, twice, and pillaged a lot of her tiles. And then I watched Paris get to about 10% of its health, and peace broke out. Both times. In my Sumeria game, the AI continent (as opposed to my continent, where I ate Rome and India) has had quite a few wars, but little in the way of results. For a while someone was defeated, but apparently they were liberated afterwards - as far as I can tell no one has any cities they shouldn't. Well, except England now, who got introduced to the joys of a colonial war declaration against them.
China also declared on me during the Spain game, but picked a bad time... about 2 turns before Cultural victory kicked in. They often amassed troops in border areas before declaring on a neighbor