As a Captain, your main allegiance is to Starfleet. However, you will also periodically interact with a variety of other Factions as part of your duties. Thanks to the temporal crisis, previously-unopposed empires and organizations are now stepping on each other's toes, and it will be up to you to mediate their disputes.
When viewing the Shuttle Mission menu, you will see icons representing all Factions with which you have a Reputation along the top. Aside from the Federation, with which you work closely already as a member of Starfleet, other Factions will regard you with reservation if not outright hostility. You an improve your Reputation with those Factions by answering their Transmissions, and successfully completing the ensuing Shuttle Missions.
Having a good relationship with a particular Faction will enable you to advocate for them when it comes tome to resolve a Dispute in an episode, and it will also make them predisposed to offer you more and better opportunities for supplies and personnel unique to their organization via their Faction Shop. If you have recruited Crew members affiliated with a particular Faction, they will likely require some of these specialized supplies to perform effectively.
In order to access a Faction Shop, tap the icon for that Faction in the Shuttle Mission menu. You will then be prompted to travel to their territory. As long as you remain in that sector, the Faction Shop will be available to you on the lower right side of your screen.