When you win a war against a opponent, you can vassalize him to be a subject-Empire. This is explained on the official wiki:
A Vassal is the most controlled type of subject-Empire. They will automatically join their Overlord's wars, aggressive and defensive, and they have no autonomy when it comes to foreign policy or diplomatic relations. A vassal also runs the risk of a full diplomatic integration by their Overlord.
However, I don't find how to do a full diplomatic integration of a vassal?
After you have been the overlord of a vassal for a certain amount of time, you can initiate the "Integrate Subject" process from the diplomacy screen.
Communicate with the vassal and select the option; it should be the bottom most option available. If you haven't been the overlord for enough time, the option will be disabled, but still visible.
When you start it, you will be shown the amount of influence required to integrate. This cost is split across a number of months, with 3 influence spent per month. This total cost varies based on the size of the vassal, with larger vassals requiring a larger amount of influence (and hence time) to be spent.
