Takeover
A takeover in BattleMaster is a game mechanic used to reassign ownership of a region to a new realm. When a realm wishes to forcefully add a region to its nation state or when a group of nobles wish to found a new realm, a takeover is used. A common abbreviation for takeover is TO (TakeOver).
Real World History
In medieval history, land was generally held in perpetuity by the family that claimed the region and could defend that claim. Land changed owners either by gift (the region's landlord giving up the title to the land) or by conquest (the "winner-takes-all" concept). In the early period, kings (or similar) tended to not have enough power to force a change in ownership by decree. By the late period, they often did, although this was usually backed up by the king having access to the largest army in the region.
Concept
There are three ways to take control of a region in BattleMaster. You can convince the local population that you are the better masters and accepting your rule would benefit them - this method is called "sympathy". The second option is to make it clear that you are the new masters and anyone daring to claim otherwise will lose his head - this method is called "oppression" or "fear". The third option is to choose a middle path, showing them that you can bring order and stability to the region, without too much bloodshed.
To control the peasants, you need at least a little bit of both. If they don't fear you at least a little, they get funny ideas. And if they don't like you at least a tiny bit, then all your oppression will just create hatred and you'll have a constant source of trouble on your hands.
Taking a region through fear and oppression is usually faster and easier, but more volatile in the long run. Taking it through sympathy is the long, reliable path. Sometimes, you don't have the time for long and reliable. Or they simply don't deserve it. Also, if they already dislike you, it is a lot easier to oppress them than to gain their sympathy.
Requirements
You can only take regions that belong to an enemy realm, which means your realm must have declared war on the region owner.
In addition, a certain minimum amount of troops is required, or the peasants will simply laugh at you. The number depends mostly on the region population. It has changed from the old Takeover system, most importantly it now refers to the total number of men from the realm starting the TO, and not the "takeover unit", a concept that no longer applies. Any unit, no matter how large or small, can initiate the takeover now.
Process
Anyone from the attacking realm can initiate the takeover once the requirements are met. Contrary to the old Takeover system, the takeover doesn't happen by itself. Once the TO has been initated, everyone from the realm that does it has an option to support the takeover in his Orders list. There is a multitude of ways to do so, all with various benefits, shortcomings and effects. Mostly, they fall on a line from most oppressive to most sympathetic.
All actions are - more or less - countering their opposite. If you try to gain fear and sympathy at the same time, you'll get nowhere. So it is important to coordinate the attackers to have them all use options of the same kind.
Each region has two threshold values, at which point you've got it. How close you are is displayed in the takeover actions screen. Note that this value only changes during the turn - all contributions by everyone are summed up and the total effect calculated at the turn change.