Owner (de facto) : indicated by colour. The de facto owner of a province has control over the military and economic resources of the province, and can concentrate armies there.
Recognized Owner : indicated by the flag above the name. Whether or not a province you control is internationally recognized has significant impacts on a lot of things. Recognized ownership of a province you don't control is grounds for war. Any change to the recognized owners of a province can be proposed by any player, and must be approved by a majority of great powers before it takes effect.
Popular Support : indicated by the flag below the name. No flag indicates popular support for the government. Popular support for countries other than the government can lead to rebellions and such.
Capital : indicated by star. Capital provinces get certain bonuses.
Industry : indicated by the number inside the factory. Shows how many factories exis in that province, which inexplicably correlates with how economically developed that province is.
Mobilization Strength : indicated by the other number. Shows how many corps can be mobilized from a province.
Occupation or Rebellion : indicated by stripes of another colour.
As an example, let's have a look at the province of Ireland.
The game has progressed a few turns, and the United Kingdom has built three factories here, as shown by the factory underneath the name. The island is still loyal to His Majesty, but there is considerable support for an independent Irish state, hence the Irish flag underneath the name. Obviously, the great powers of Europe still consider Ireland to be a British possession, so the Union Jack is above the name. In the event of mobilization, the province will provide two corps of loyal British troops.