**Handle with care - see comments:**
Imagine some system with states and transitions between them, e.g. an automaton as in computation theory or a Markov chain as in stochastics, for simplicity assume that you have an assigned initial state. Then you can label the transitions by letters and record your transition history, starting from the initial state, by forming words out of them. This gives you a semigroup. Now declare two words to be equal if they lead you from the initial state to the same state (careful - this is not the way you usually use automata in group theory). Then left reversibility means that you can, from two given states always go on to reach one same state, i.e. any two starts of your program can lead to the same outcome.

A non-deterministic terminating algorithm gives a meaningful example of such a thing. Non-deterministic means that there actually are several ways through your state diagram, terminating means that you always end in the final state.

Another (boring) example: The multiplicative semigroup of a ring - the zero always does the job. This is like an automaton which always has a one-step way from any state to the unique final state.