This is the ability to sway a system to favor your side. The actions you take in the game can affect loyalty and ultimately control of systems. For example, if you use diplomacy on one system in a sector that has several neutral planets and manage to sway that system to your side, you may also win over a few other neutral planets in that sector whose loyalty was already greater than 50% for your side Similarly, if you take over planets by force, the planet may go into uprising because they did not support you enough. If you blow up one of your planets with the Death Star in a sector where planets you control don't fully support you, you may send several of your planets into uprising and may even lose a few of them. In general, events that happen on one system tend to affect the other systems in that sector to some degree. Some events, like destroying an entire sector, can have a global effect on all core sectors. Rim systems do not seem to have their loyalty affected by events elsewhere in the galaxy.
Often, control is only maintained on a planet because the controller is able to meet the garrison requirements to prevent it from going into uprising.
 

This is the ownership of a system regardless of which side the population favors. It is entirely possible that the Empire controls a planet, but that it fully supports the Alliance.

This is basically the popular support for one side or the other. It is indicated by a red/green bar underneath each planet, in which green indicates
support for the Galactic Empire and red indicates support for the Tebel Alliance. Note: In order to control an uninhabited rim planet, you must move at least one trooper regiment off of a fleet around that system to the system itself (support will then be 100% for your side). You will notice, though, that the Garrison Requirement says it is zero. However, if you move that troop off before you develop the planet, you will lose control of that planet. Once a facility is built on that planet, you will have full loyalty and can move the troop on if necessary.