Castaway Cay is a private island in the Bahamas owned by Disney Cruise Line. A variety of Disney's Bahamas and Caribbean cruises include a stop at Castaway Cay. Departures are from Port Canaveral (Florida), which is the port closest to Orlando.
Ships park at a big dock, which means the ship is conveniently accessible throughout the day. There is transportation around the island, including a tram to Serenity Bay - the beach exclusively reserved for adults.
Disney provides this excellent video about a day at Castaway Cay.