The campaign missions are started from the "co-op" tab.
In the co-op tab, there is a drop-down menu to select from the campaign missions. The missions with a faction logo on them are campaign missions from the original game (before Forged Alliance). There are Forged Alliance campaign missions. There are also "new" campaign missions created for FAF.
You can launch a co-op lobby from the co-op tab. Obviously, when you launch the lobby, you're the only human in it.
From the lobby, you can edit the game's options (increase or decrease the difficulty, or other options).
You can launch the game from the lobby even if you're the only human player. You don't need anyone else in order to play a "co-op" game. You can let people join your lobby, or you can put a password on it to keep them out.