Missions can be obtained by gate guards in the same way signet can be. However, you can only receive missions from your home city, not outposts or consulates like a signet. Also Missions are done for Rank. Rank differs from fame in that fame allows you access to more quests, wherein rank allows you to obtain high priced items with your conquest points and other perks, such as an Airship pass.
Missions are for and by your city, and only your city. Windurstians can’t just walk into Bastok and expect to be offered missions. But they can require travel to these countries and sometimes cooperation from the populace in order to complete your Mission. They can vary from tracking down a certain someone to ensure their safe passage through dangerous lands, to quelling a climbing beastman population, and even becoming a detective and solving a mystery or two. Each mission is progressively harder and the rewards, gil and more importantly points to your rank, are better as well.
Supply Runs can be made to outposts located in regions you country controls that will give you points toward you rank. These offer the least amount of points, but are fairly simple and easy to do. Expeditionary Forces can also be sent out from your city. All you need is to party with members of level 25 or more. These parties will be sent out to an uncontrolled area to reestablish your countries presence and lower beatsman influence.