In a co-op campaign, you each get your own families (so you can make your own choices about your tech trees and campaign characters), but your families are automatically entered into an alliance. The win condition is either one of you becoming Shogun, and we were assured that there is no backstabbing allowed in this alliance. It sounds virtually the same as the single-player campaign mode except you're working together to win the game instead of just you on your own.
Shogun 2 takes place in 16th Century Japan during the Warring States Period where various clans were warring for power. This eventually ended when the Tokugawa clan won. In Shogun 2, you have the chance to change history. In the campaign, you pick a family which you'll be playing throughout; furthermore, you have a lot of different customization choices you can make for your chosen family. You can make political decisions for your clan (e.g. proposing marriages to seal alliances with other clans). You can choose whether your clan specializes in the Way of Chi or Bushido (these are like tech trees). You even get characters on the campaign map that you can level up and specialize as you see fit. These characters can be generals (which are used for the RTS combat which I'll get to in a second), government policemen, and yes, ninjas that you can use to assassinate other characters. There's a huge amount of RPG-like elements incorporated into the campaign to give the game a real sense of telling a story.
Huge battles make a return as well during the RTS phases.
This page contains co-op info for Total War: Shogun 2 on the PC. All information about Total War: Shogun 2 was correct at the time of posting. Information is subject to change. If you see any errors please email us.