An important example from CS would be the semigroup action on the set of states of a deterministic finite automaton.

The action takes a state from Q, and a symbol from E, and returns a new state. ie f:QxE->Q. In the case when you have some "no op" action the semigroup action is actually a monoid action.

This all corresponds to the DFA moving through states as it "eats" the string.