An area of the board (empty or not) has been captured by a player if all of the opponent's pieces in that area are dead, and for any possible move by the opponent in the area, the player has a counter-strategy that kills all of the opponent's pieces in that area.
It is never advantageous for a player to move in an area that has been captured by the opponent. A captured area may as well be assumed to have been filled with the capturing player's pieces, as this does not change the strategic value of the position.
The most common example of capture is the second row edge template
The two cells marked "*" are captured by Red. If Blue plays at b5, Red can play at c5, killing b5. Conversely, if Blue plays at c5, Red can play at b5, killing c5. Therefore, both cells are captured and the above position is strategically equivalent to the following.
For a more complex example, consider the following position, with Blue to move:
The two cells marked "*" are captured by Red, because if Blue plays at either one of them, Red can play the other, killing Blue's piece. Since each Red-captured cell can be treated as a red piece, it follows that Red is connected to the bottom edge by edge template V2a, even though Red does not have an actual piece at f3.
Here are some other examples of cells captured by Red:
Captured cells and dead cells
Any cell in which a player actually has a piece is trivially captured by that player. Moreover, since dead cells can be treated as cells of either color, an empty dead cell is captured by both players. (Dead cells containing an opponent's piece may also sometimes be captured, but when considering such cells as part of a captured area, beware of the interaction between multiple dead cells).
The analysis of dead cells and captured cells may sometimes go through multiple iterations: as some cells are discovered to be captured, they create other dead cells, which in turns may create additional captured cells, and so on.
For example, consider the effect of a red piece at b2:
First, b2 captures the two cells marked "*". Then, because the cells marked "*" can be treated as if they were red pieces, the cells marked "+" become dead, and therefore also captured. Thus, a single red piece at b2 has captured four other cells.
Moreover, if there is an additional blue piece at a4, Red b2 actually captures five cells:
First, b1 and c1 are Red-captured and a1 and a2 are dead as in the previous example. Finally, since a2 can be treated as a red piece, it also kills a3.
Captured is not the same as connected
Based on the example of the 2nd row template above, one may wonder whether cells that are part of a template are automatically captured. This is not the case. To see why not, consider an interior bridge template:
The two cells marked "*" form part of a bridge, but they are not captured. Indeed, if Blue intrudes into the bridge at d3, Red will lose because she cannot simultaneously defend the bridge and prevent Blue from connecting at d4. On the other hand, had the cells marked "*" been red pieces, the position would have been winning for Red.
Henderson and Hayward, "Captured-reversible moves and star decomposition domination in Hex".