Suns
Suns produce the energy that pays for tech. They are drawn as a filled disc in their owner's colour; the faint tinted circle behind one is its radius. A base draws the same kind of circle and grants the same tech.
Taking a sun
Drag from one of your planets to a sun and choose how many probes to send. Every 100 probes becomes 1 super-probe on arrival, and the send is trimmed to a whole number of them, so an attack needs at least 100 probes. The super-probes fight the sun's garrison; win and the sun is yours, along with its energy income.
Two things move that price. Launching from inside the radius of a friendly sun or base, the Conversion tech makes a super-probe cheaper. Attacking across a galaxy border makes it dearer, up to 1,000 probes right after the border opens (see Attacking & Capture).
Holding one
Suns make no super-probes of their own, so a sun's garrison is only ever what you sent it. Every attack wears it down, and a contested sun changes hands after repeated attacks. Taking one always needs more super-probes than it holds, whatever your Assault level: a sun fights without tech on either side.
Sun swarms
Drag from one of your suns to another to relay super-probes between them, which is how you keep a front supplied. They travel like ordinary swarms, at a speed set by where the source sun sits in its galaxy, and they fight on arrival. Aimed across a border they pay the same toll everything else does.