It's not even "e-mail chess", because it requires another direct player, and if I remember well (someone correct me if I'm wrong), there were only single player missions sharing stuff of the other players.
Essentially the main purpose was to fight against enemies ("nemesis") of the other players, or to have some kind of "arenas" (to simplify the thing) where to upgrade your character outside of the main story campaign, since the main character was shared between all those modes.

Maybe there was also some kind of leaderboard online too, and I suppose most people feels it important to test themselves, like having achievements.

Except for a gear that was only obtainable in one of those modes, all the other elements are present in the main campaign too.

I think I'm beginning to understand. So it's not as passive as I assumed. I thought you would raid someone's outpost and the player would answer back, but the raids happen in real-time with both players interacting then?

There is no interaction at all with other players, for example you can "avenge" them by killing an ork who killed them, but nothing will change in their game, it's just an enemy that was uploaded to the server to be used as a target for other people... you will only know the nick who you avenged but that's it.
The same for the other modes, just shared stuff from other players, but without interaction between the two play sessions.