Expiry is when your residence visa becomes invalid (note that expiry is NOT the same as cancellation - residence visas are NOT automatically cancelled when they expire, but they do become invalid). This happens after the expiry date (usually 3 years after it was issued, sometimes 1 year), or if you are outside the UAE for 180 days or 6 months (assume 180 days to be safe). Expiry does not mean your residence visa is cancelled, as some expats have found out when attempting to re-enter the UAE on a visit visa and failing because a previous residence visa wasn't cancelled.
Cancellation of your residence visa is an active process which needs to be done even if the visa has expired. Otherwise a new visa, whether residence or visit, cannot be issued. If you are overseas, then cancellation can still be done but your sponsor needs to go to the residency department on your behalf with at least a copy of your passport, and possibly the original.