I have two favorite games - Half-Life and Quake 2. They are the games that introduced me properly to the First Person Shooter genre. Half-Life with its immersive single player experience and Quake 2 with its fast paced multiplayer.
Half-Life delivers a great single player experience with a focus on story and a very transparent hero, Gordon Freeman. He is someone that you can relate to and feel comfortable taking the role as. There is also a very mysterious character, always in the background, named G-MAN. And that's another thing that Half-Life features - interesting characters. I find the level design to be very good, with proper guidance and always a motivation to push forward. The newer Half-Life games feature a continuation of good single player experiences. Something I will always remember from Half-Life is the introduction level, where you are riding a train inside Black Mesa - an effective way to introduce the player to the Half-Life universe.
The Quake games offer great multiplayer. They have a focus on momentum. Speed, flow, accuracy, reaction, prediction and patience are just a few of many of the ingredients found in a Quake game, which makes it so interesting. There are so many ways to play the game. Many levels offer really good level design, like the classical map Campgrounds or a newer map like Trinity that was introduced in Quake Live. Many levels also show great visuals without being too busy with details, something that often act as a distraction in multiplayer games.
Both of these games, Half-Life and Quake2, are the main reason why I started doing level design. I love them and I wanted more.