[thejeff]

They will join together to fight in one battle, unless the enemy army in C attempts to attack either A or B. In that case, one of two things are likely to happen. Assuming C is attacking A, either A & C will fight in A while B invades province C, or both A & B will fight C in province C.
All movement is resolved then battles take place where ever hostile armies end up together. This is why you'll never catch an enemy army chasing it, you have to cut it off.
When armies cross each other (A attacks C and C attacks A) one army moves and the other stops. Very rarely the armies miss each other and swap places or neither moves.


Edit: Hah, my answer was ninja'd but I anticipated and answered the next question as well.