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Conquering a region is done by initiating a takeover and then supporting it with takeover actions.
There are many ways to conquer a region, but they all rely on one of two basic principles.
You can either make the population so afraid of you that they give up hope and submit to your rule. This is the easy way, but it means constant vigilance for the future because once you have oppressed a people, you need to go on oppressing them or they will remember all the bad things you have done. Regions taken and ruled by fear are more easily lost to apathy or other invaders.
You can also work with the people, improve their lot in life, bring them whatever cultural or technological marvels your realm calls its own, and convince them that a better future lies with you. This is called a takeover by sympathy and is a longer and more difficult process, but once you have succeeded, the people accept your rule willingly and will be less easily taken in by others.
Note that all things have opposites. You can not rule entirely without fear, as loyalty without fear breeds contempt. You can also not rule entirely without sympathy, becauce enough hatred will overcome fear.
Therefore, there is also a third, middle path: The balanced takeover. This is the most challenging of all, but if done right will result in a stable, easily controlled region that needs very little additional attention. A balanced takeover is easily described: A balance of fear and love - but the most difficult to accomplish. The exact balance between fear and sympathy varies from region to region and depends on many factors.
Everyone from the realm that runs the takeover can contribute with his soldiers. Allies can secure the area, but not support the takeover itself.
While there is a multitude of actions available to support the takeover, they are listed in the game from the most fear-generating to the most sympathy-generating. The more extreme options generally work on the principle of opposites - those with the most impact on fear will also reduce sympathy and vice versa. The options in the middle are more balanced, with a positive effect on both, but less pronounced.
Takeovers are stopped if at the turn the number of soldiers from the realm doing the takeover has fallen below half of the number that would be required to start a takeover (this can be slightly different from the number that was actually needed, due to changes in population, etc).
So attacking and defeating the conquerors is how you stop a takeover, because evading, retreated, scattered or rallying troops do not count towards this requirement.
In addition, various actions can slow a takeover. Since takeover actions require men and hours, anything that reduces those will slow down a takeover. Infiltrators can incapacitate nobles, making them unable to partake in the takeover. Battles, even if lost, will cost the conqueror men and hours. Some activities that influence region stats like sympathy will also influence the takeover.