The mop up setting is used to designate a pitcher be used in blowouts (either winning or losing). Blowout is determined by the deficit (or lead), the inning, and the ballpark. e.g. an 8 run deficit in the 2nd inning would not considered to be a blowout, but the same deficit in the 8th inning would be. And 8 runs means something different in Petco versus Coors Field.
Mop-up designated pitchers can possibly be used in non-mop up situations. They will always be used before pitchers marked as 'rest'. For example, if you have 5 relievers and 1 is marked as rest, 1 is marked as mop up, 1 is marked as long relief, 1 is marked as setup and the last as your closer, and your game went to extra innings, if your closer, setup, and long relief had already been used, then your mop up pitcher would see action.
It's good advice to give your mopup pitcher a high target and max pitch count so they can truly eat innings. It's also recommended they have a lower than normal Auto Rest setting so they can be used even if they are a bit tired. This allows your team to save its better pitchers for optimal situations. However, don't forget to consider the Manager Setting for mopup usage as you don't want tired mopup pitchers squandering big leads.