A bunch of teens travel to an isolated island for a secret party. (BAD IDEA BAD IDEA NEVER DO THIS. SECRET PARTIES ON ISLANDS NEVER END WELL.) Anyway, one by one they start dying. And one by one their options for getting help are eliminated. You begin to think it might be one of them, but it CAN'T be one of them, but no one else is on the island, but someone else MUST be on the island. Or maybe it's a ghost.
I have never read the original Agatha Christie novel but I DID read this amazing book called Weekend by Christopher Pike that has the same basic premise (except the killer contaminates the water supply, so the teens are getting murdered AND they are barfing and sweating and other terrible things.) Oh man.