To capture an enemy base you must first kill off everything within it, then get a unit able to perform the capture into it. Whichever unit you choose for this, will be instantly returned to full health.
Any military unit can capture a base except aircraft, transport ships, and probe teams. If the base is a sea base, then you must use a unit able to cross water, be it an attack ship, isle of the deep, or a land unit with the amphibious option built in.
When a human base is captured by a human player, or an alien base is captured by an alien player, it losses one of its population. When an alien base is captured by a human, or a human base is captured by an alien, it looses all but one of its population, and will produce various numbers of colony pods for its fleeing population, including sea colony pods if near the water.
You can only use probe teams to do a very costly mind control capture of a base.
Bases that have drone riots for several turns, have a random chance of joining the Free Drone faction, if they are in the game.
You can also intimidate others at times into hanging over a base to you, freely, nothing traded.