Territory
A player's territory usually refers to the part of the board that the player controls or has some influence over. One way to define territory is to say that it is the set of hexes occupied by the player, plus any empty hexes adjacent to them.
However, it is often better to consider the idea of territory in a looser sense. We can say that a player has influence in an area of the board if the player has many potential ways of connecting there. Sometimes even a single stone that doesn't look very useful can increase a player's influence. If a player gains such a stone, we say that the player gained some territory.
Example
Consider the following situation, with Red to move.
Red could simply connect to the edge by playing at "a". However, it is often better for Red to play "b" instead, as this will still connect Red to the edge, but gain some additional territory (namely, more possibilities to connect towards the top).
Gaining territory by intrusions
Often a player can gain some useful territory by intruding into the opponent's templates.
