I assume you’re talking about Europe. The more northerly Mediterranean resorts wind down towards the end of September (and in fact some become decidedly chilly and unsettled around this time). Resorts on Greek islands such as Corfu, Thassos and Lefkas usually close up by the beginning of October (though you will still find restaurants and bars open in the larger towns). The Balearics (Mallorca, Minorca and Ibiza) are similar. You may find some resorts in southern Spain and Porugal going on a little longer but you will take a chance with the weather. I've been to some of these places as the season draws to a close and if the weather has broken they can be a bit depressing with the bars and restaurants gradually closing up.
October is usually OK in Cyprus, Malta, Turkey and the southern Greek islands (Crete, Rhodes) but for a guaranteed good time weather and entertainment wise head for the Canaries. They operate all year round and in October they will still enjoy warm and maybe even hot weather and you will find almost all the facilities open.