Guild missions are weekly challenges for your guild. The currently available missions are listed in the Guild Missions tab of the Guild panel. Select the mission you want to run and start it; you'll then have a limited amount of time to complete its objectives. If you fail, you can run it again at no cost. The first time your guild successfully completes a mission each week, it will award your guild favor. The first time you personally complete each mission in a week, you'll receive a personal reward.