First of all a priest never stops being a priest - he is “a priest forever”. Still, priests do, at some point, cease to be involved in full time ministry - in other words, “retire”. This means that they are no longer a chaplain, parish priest (pastor), teacher, or whatever it was that they were doing beforehand. This is true equally of both diocesan priests and those who are members of religious orders. While some dioceses do have a set age at which they prefer their clergy to retire from full-time ministry, this is not normally a hard and fast rule - much depends on how the priest is doing health-wise and the needs of the dioceses. So sometimes a priest can ask (or be asked) to postpone retirement for a few years (and perhaps accept an assistant, a seminarian, or less-demanding appointment). It would be a harsh bishop who forced a priest into retirement against their will (admittedly some priests do need persuading).
Once they are no longer in full time ministry, such priests often continue to serve their diocese’s needs, for example though doing supply work (covering for another priest who is sick of away) or by living in a parish and helping the parish priest (very useful at Christmas and Easter). Sometimes a priest can also find a way “around” retiring such as by being a live-in chaplain at a rest home or retirement village.:rolleyes:
Incidentally, the same is often true of bishops who are required to submit their resignation at 75. Depending on their state of health and the state of their diocese (as well as other things), this resignation way or may not be accepted straight away. Once retired, they remain a bishop and often help out in their own diocese (or another) with confirmations and other such work.