Conquest, Tax, and Rebellion
Conquest is how you turn a military win into a lasting economic advantage. Beating an enemy in the field is satisfying, but conquering their colony keeps paying you tick after tick. It is one of the strongest engines of growth in the game, and one of the hardest gains to hold onto.
Conquering a colony
To conquer a colony, you defeat the units defending it in battle. Conquest does not destroy the colony or its owner. The colony keeps existing and the owner keeps playing, but it now answers to you. A colony that is conquered cannot conquer others while it is held, so chaining a victim into a tool for further expansion is not possible: each conquest stands on its own.
The tax you collect
Once you hold a colony, it pays you tax every tick. Tax scales with the size of the conquered colony, so a large, developed colony is a far richer prize than a small one. Importantly, the tax is added to your income without being subtracted from the conquered colony. The victim is not bled dry resource for resource; the value comes to you on top. This makes a healthy conquered colony something you actually want to keep alive and growing, not strip and abandon.
Rebellion: taking it back
Being conquered is not permanent. The original owner accrues control ticks the whole time their colony is held. After about 72 control ticks, they can start a rebellion to reclaim it. A rebellion takes roughly 24 ticks to complete. During that window the rebelling player has to stay strong: if they lose another colony defense while the rebellion is running, the rebellion fails or resets and the clock effectively starts over. So conquest is a contest of staying power on both sides. The conqueror wants a long, quiet hold; the victim wants to survive long enough to throw it off.
Liberation
There is one more way a conquered colony goes free. If you, the conqueror, are yourself conquered by someone else, the colonies you were holding are liberated. Your subjects are released the moment your own colony falls. This makes overextension dangerous: the more colonies you hold, the more you lose in a single defeat, and the bigger a target you become for anyone who wants to free your victims at a stroke.
See also
Read Combat: How Battles Are Decided for the battles that lead to a conquest, Outposts and Camps for the other way you hold ground, and Economy and Income for how conquest tax fits into your overall production.