Basically, supply is a gameplay mechanism that makes immobile structures only work in its player's territory.
It's important to note that both AI and Human players use this mechanism.
A planet can be either supplied or not; there is no numerical value. A planet is supplied if that planet or at least one adjacent planet has a Command Station controlled by the friendly team. To put it in another way, each command station provides supply for its team on its planet and on all adjacent planets.
Supply is shared among the team: all human players have supply around their command stations, and so do the AI players.
Maintaining supply on a lone planet is more difficult than on a frontier planet: when the command station is destroyed, the frontier planet has adjacent allied planets to maintain supply on it; a remote planet doesn't.
Supply is required for nearly all immobile structures.
However, mobile units don't require supply: all combat ships can fight in enemy territory.
It is to be noted that enemy planets adjacent to allied planets have supply that can be used. With a Mobile Builder, turrets and other structures can be built on adjacent enemy planets; that strategy is called beachheading.
If the planet (the one in the background) is destroyed, by the detonation of a nuke or the act of a Mining Golem, for instance, it cannot receive nor provide supply. A command station can still be built, but nothing else.
A Co-Processor also prevent supply on their planet, but they don't prevent supply providing. For instance, a Human command station on a planet with a co-processor won't supply its own planet but will supply all the adjacent planets as usual.