A province in Imperialism is an area of (several, and usually a dozen or more) land tiles containing in one tile a named city, town, village, or hamlet with a military encampment. In the Europe scenarios they are based on real regions, though the location of some towns such as Belfast may be a little astray.
Province boundaries, shown as white wriggly lines on the map, are unchangeable. Each province is part (or the whole) of a nation. At the start of a standard game, each nation has one capital city province, each of its other provinces containing only a hamlet. The hamlets in minor nations tend to have three Minutemen or Militia guarding them, while the standard in a great power is four.
A province may be captured by ground units belonging to a great power attacking from an adjoining province or across an established beachhead. It may then be retaken by friendly forces sent in the same turn, but if two allies both attack a province in the same turn only one can succeed, the other having its involvement shown as "Battle Preempted" (and there is no guarantee that the stronger force will succeed, especially if it is using a beachhead while the smaller allied force comes by land).