This is the first tabletop roleplay game to be featured on this list, and it's a good'un! Originally released in 1991, the Vampire the Masquerade roleplaying game was the first of quite a number of games to be set in the World of Darkness, a universe created by White Wolf Games.
Players assume the role of Vampires who are trying to survive in a modern setting. Vampiric society has split into two distinctive sects -- the Camarilla (who are for integration and "masquerading" as normal humans) and the Sabbat (who just want to let it all out and prey on humans). Whether you are playing as the "goodies" or "baddies", there are always things more powerful and bloodthirsty than you to defeat.
Developed by Troika Games and published by Activision,Vampire: The Masquerade - Bloodlines was released in 2004. A lot of the roleplay aspects of the game follow the ideas set out in the original tabletop adventure, and a lot of the story depends on which clan players assign their characters to. While this game has a definite storyline, you can still exercise your need for roleplaying by choosing a clan and making decisions to determine how this story is completed.
Even though it is out of print, the Vampire: The Masquerade is still available second-hand for various prices via Amazon or Ebay. For a game that was released a pretty long time ago, it is still going strong on Steam at $15 -- and has a dedicated modding community to boot.