Yes - just a normal cruise that "repositions" the ship in a new port for a new season of cruising in a different area. You would have to fly home from the final destination. What's on offer will depend on the size of the ship and the facilities aboard, but most cruise ships have a programme of activities that can keep you busy all day and on into the night.
I have never cruised alone but my impression of single cruisers is that they have to be pretty happy in their own company. The various activities would give you an informal way of meeting people though many passengers are couples and/or travel in groups. People are friendly and you can always find dinner companions, but as a single traveler, you need to be pro-active to find the company you want.