In the diplomacy screen, on THEIR SIDE, in the middle there is an option attack. It's in the same area as the treaties, alliance option, etc. This will give you a drop-down menu of all races they are willing to attack (meaning their relations with the target are neutral or worse). Once they are at war (whether you started it or not) there is an option to make peace.

The attack option exists on your side, but that means YOU are declaring war on the selected civ. If you're going to attack someone anyway, it's a good way to make a bit of cash out of it.

Also if you are economically sound like I always am you can just buy a race to start a war with another race. Plus the higher your diplomatic rating the easier it is to get them to start a war and the less it costs. Sometimes just giving them a small puny tech or a weapon tech will get them started. I once had the entire galaxy at war with one another and I was friends with them all reaping the rewards as I would jump into them when a race had lost significant holdings and ships. lol It's kind of exploiting the ai to do this though, but, is still something fun to do from time to time.

...and that's when you pay someone else to attack them.