The quickcast marker allows your mage to perform 1 quickcast per round in addition to your normal action(s). This can be done during the First Quickcast phase, during the Last Quickcast phase, immediately before or immediately after any friendly creature's action.
During your normal action you can perform a full action, perform a quick action, or move and then perform a quick action. A quick action can be a quickcast spell, but regardless, you only flip over your quickcast marker if you are doing the special additional quickcast that only your mage can do; you can still do a quickcast during your action phase if you did your first quickcast, and you can still do your final quickcast if you didn't do your first quickcast and didn't use it during the actions phase.