When you join an Alliance you will gain access to Alliance based Missions. Every member in your Alliance will be able to contribute to the goal of the Mission and earn rewards upon completion, given that you became a member of that Alliance before the Mission event has begun.
Keep in mind that if you leave your Alliance you will lose access to the associated Alliance Missions. Additionally, if you join an Alliance after a Mission has already begun, you will not be able to participate in that Alliance Mission.
You can view the progress of your Alliance's Missions in the Missions tab or within your Alliance tab.