That is to say, if you bring a fleet to a world, and there is no other player with fleets at that world, and there are no I-Ships or P-Ships on that world, then you will become the new owner of the world, as well as of any (empty) fleet keys at that world.
On the other hand, if two different players bring their fleets to a world at the same time, then ownership of the world does not automatically change (though it may go neutral if a player subsequently fires at home fleets; see Air-to-Ground fire). Combat aside, if the world was previously owned by anyone, it will remain owned by that player. If the world was previously neutral, it will remain neutral. If there are any empty fleet keys at the world, they will become neutral (even if they were previously owned by a player).
This means that if you want to prevent your worlds from being captured by the first player to bring a fleet there, you must leave at least one I-Ship or P-Ship at the world. The presence of this ship will count as a ship in your name, and will also serve to keep ownership of the world in your name as well (unless the other player fires at your ship and destroys it, but that will take an additional turn).
Any fleets that you declare to be At Peace don't count towards this rule. If you bring a fleet "at peace" to a world at the same time another player brings a fleet not "at peace" to the world, the other player will capture the world.
Also, you will not capture worlds from your allies, even if you are the only player there and your fleets are not "at peace". But having a fleet not "at peace" at an ally's world will serve to prevent another player from capturing the world.
Note that if you have any empty fleet keys at a world, you will lose ownership of them if another player brings a fleet there, even an ally. At the best, they will become neutral; and at the worst, they will be owned by the newcomer. If you want to keep ownership of your fleet keys, it is best to keep at least one ship on each one.
Also note that Commandeering and gifts will change ownership of a fleet, but since both of these occur after the world-capture test is made, for the purposes of capturing the world the fleet is considered owned by its previous owner. The world will not change hands as a result of a fleet gift or Raider capture until the next turn.
No one can own a world that has zero population. If you need to capture a world with no population, you will have to migrate people there from a neighboring world--unless you are lucky enough to be a Positronix, in which case you can simply drop a robot or two from one of your fleets.
If the world's population is robotic, only a Positronix can own the world. So even if you are the only player to bring a fleet to a Positronix's robotic world, unless you are also a Positronix, you will not capture the world from him or her.
If the world's population is 100% converts, you cannot capture the world unless you first eliminate some of the converts, for instance by unloading consumer goods.