Each time you complete a space mission you receive credits and experience as you would from a quest. Additionally there are many quests associated with space mission. These quests reward fleet commendations (and some higher level quests also reward other types of commendations). Each space mission will have a quest to complete it for the first time. As you complete those quests you will unlock repeatable quests. The repeatable quests will require you to complete one or 2 specific missions. Most of the repeatable quests can be completed daily except for the heroic quests which can be completed weekly (I.e. a different heroic space mission quest for certain days of the week). Some space mission quests will reward you with items such as ship upgrades, but most of the rewards from space missions will come from purchasing items using fleet commendations.
The fleet commendation vendor is on the fleet station in the section opposite the trade network terminals. For Empire players the fleet commendations vendor is S-1A0 and for republic players it is S-4S0. Next to that vendor is a vendor that sells ship upgrades for credits.
The fleet commendations vendor sells certain special ship upgrades for flet commendations. These include Power Conversion Module, Electronic Warfare Pod, EMP Generator and Proton Torpedo Launcher.
The commendations vendor also sells a variety of crafting boxes and storage boxes. The crafting boxes contain random crafting materials while the storage boxes contain a random item. In each case there are different types of boxes costing different amounts of fleet commendations. The more expensive ones contain higher quality materials / items. Generally the contents of the boxes can be either useful or can be sold for a decent price on the trade network so they're a good way to spend surplus commendations.
Lastly, the commendations vender sells social gear. Several outfits are available with 2 being available to both imperial and republic players, while each side also has a 3rd, faction specific outfit.