For example, for a Boys Run a boy will be "running around a girl". The boy walks in the semi-circular arc, into the spot where the girl was standing, and ends facing the opposite way from when he started. The girl slides over into the spot where the boy was standing, and remains facing the same wall the whole time.
For a Girls Run a girl will be running around a boy. The girl flips over to face the opposite wall, while the boy slides over without turning.
The most important thing to notice is that one dancer turns around and the other doesn't. Indeed, this is exactly what makes the action of Run different from the action of Trade, done by the same two people. If they were doing a Trade, they would both turn around. For a Run, only the dancer who was named turns around.
Note that the way the dancers are designated for a Run is different from almost all other calls. For most calls, the caller names the group of people who are going to do the action. The dancers who are not mentioned don't participate in the call. For example:
Run is different. It doesn't fit that pattern. For a Run only one of the two dancers who is going to be involved in the action is named. The other is implied. The other dancer will always be one that does not fit the description, the "opposite" of the type named. For example:
In a line of four, there are typically two dancers of each type, for example, two boys and two girls. In that case, the run action takes place in each half of the line. For example, for Boys Run, each boy will be running around the girl in his own half of the line.
The same is true in a wave of four. (A wave is a kind of line.) From a wave with the girls in the center, Girls Run means that each girl will be running around the boy in her own half of the wave.
Both "runners" might be moving in the same direction, or they might be moving in opposite directions -- whichever is necessary to run around another dancer in the same half of the line. For example, for a Centers Run from a one-faced line, one runner will be flipping over to the left and the other will be flipping over to the right. Similarly, one of the ends will be sliding right while the other will be sliding left.
If the Ends Run, they will be "running around the centers", which means that the centers will need to slide outward to get out of their way.
Note also that if the two dancers start as a couple, facing the same direction, they will end in a mini-wave, facing opposite directions. If they start as a mini-wave, after one runs around the other they will be facing the same direction, as a couple. One of the most common mistakes that beginners make with Run is to try to "fix" this by turning around to make the formation the same as it was before. The formation is supposed to change. It's OK. Don't try to fix it!