War types:
Skirmish: 25 turns to do whatever the heck you want, small prestige cost
Retaliation War: Retake a city someone took from you, set number of turns, has a prestige cost and ends when you retake city and you get a bonus if you successfully take it.
Annihilation War: Unlocked from act 1 by having a leader trait, and then later from a government type, war goes on infinitely until one side is destroyed or both sides agree to a Negotiated Withdrawal. Has a massive prestige cost that scales up depending on.. their population? your score? not sure, but it can be like 1.5k+ after early game, and provides a tiny bonus for winning.
Holy War: Take a province of your own religion that another player has. Small prestige cost depending on number turns you set CB for, small prestige bonus for winning.
War of Expansion: Choose a city near your own cities, assimilate or raze it. War is for 25-55 turns, war is immediately ended when that city is taken.
I think those are the only 5, maybe another one gets unlocked in late game, I don't remember.
Realistically, you always choose Skirmish unless you think turn timer is too short if you are confident you can win. You choose War of Expansion/War of Retaliation if you think you can beat an opponent in their current state, but are wary that they could bring forces to bear against you in like 10 turns (so for example you know of their 1000 army strength, 900 is currently 40 tiles away in another players territory in a war, you think you can march across the border and siege down a city before they can muster new troops and repel you.
