As of right now i pretty much only play either League of Legends and Counter-Strike: GO, but some of my other favorite games are the Half-Life and Left 4 Dead series.
Half-Life series i just think is the best single player game ever made, the game play is interesting and fun, the weapons are imaginary and original, the story is amazing and great, and i really like all of the soundtracks.
Left 4 Dead was just really fun to play with friends when it first came out, i think we spent close to 600 hours on it, good times.
Counter-Strike i have been playing like ever since the 1.4 or 1.5 i think, and it just has been thousands of hours of rage, joy and memories.
League of Legends is currently the game i spend most time playing, the basic idea of the game is great, 5 versus 5, the more skillful you are the more you will enjoy it, but you can still enjoy it even though you might not be good at it.