The Capital of a nation is the province which holds the seat of government. The Capital province is the target for several spy missions such as Infiltrate administration and therefore benefits from buildings which confer a high spy defence. The capital is also used for decisions such as Institute Liberum Veto and Formalize Scales, Weights and Measures.. Population growth is also higher in the capital.
Taking the Capital
In a peace settlement taking the Capital of a nation requires certain specific conditions. In the case of an inland Capital it needs to be isolated from all other provinces. In the case of a sea province then the opposition nation can have no other ports.
Moving the Capital
Moving the Capital to a new province costs 1000 and reduces Stability by 4 points. To move to the new world, in the case of colonial empires, such as Portugal the move has to be made in stages. Firstly the Capital has to be moved to an island, such as the Azores or Maderia and then secondly to the New World province of choice. If you are a member of the HRE, you cannot move your capital to a non HRE province, which means you first have to add the province to the Empire before you can move its capital. This means you cannot move your capital to the new world if you are a part of the HRE. One tactic to get around this is to leave the HRE, remove your capital province from the HRE and then move your capital wherever you like. Therefore, choose carefully which you need.