A multiple threat is when a player threatens to connect in two or more different ways. Ideally, the opponent cannot defend against all of them simultaneously; or at least the presence of multiple threats severely constrains the opponent's options.
Whenever possible, a player should make each move achieve at least two different goals. Moves that contain only a single threat are generally not hard to meet. Moves that contain multiple threats are more difficult, and sometimes impossible, to stop.
Connection by double threat
The most common example of a multiple treat is a double threat connecting two of a player's groups. In the following position, Red's two groups are connected by double threat at the two cells marked "*".
If Blue moves at one of the cells marked "*" or its neighbors, Red can respond at the other one, thus guaranteeing a connection between the two groups.
Connection by triple threat
Consider edge template IV1a.
The red stone is connected to the edge by (at least) three threats at a, b, and c. If Red plays at a or c, Red is connected by a ziggurat:
If Red plays at b, Red is connected by edge template III1b:
Since there is no hex in the overlap of all three threats, Blue cannot block Red's connection.
In the following situation, Red can connect the two groups by moving at any one of the cells marked "*".
However, these threats are overlapping; each of the three threatened connections passes through the cell c3. Therefore, by moving at c3, Blue can neutralize all three threats at the same time, denying Red the connection.