With games, genre usually refers to how the game plays(Fighting, Strategy, Action, Adventure, Shooter, RPG, Puzzle, Racing, Sports, etc.), unlike movies which is more about story/setting. Some games ride the line between different genres as well.
Personally my favorite game genre is a bit neglected. Not enough companies really make turn-based strategy games any more. I do still need to get X-Com: Enemy Unknown since I always loved the original game in the series, but other then that the last big turn-based strategy release was from the same company with Civilization 5. There are some smaller releases in the genre, but they are still few and far between while all the developers at major companies pump out shooters, open world action games, and MMOs that inevitably go free to play because they can't sustain a subscription fanbase. Or so it seems like most companies are focusing on these days. Don't get me wrong, I like a wide variety of genres, but I get the most enjoyment out of a good turn-based strategy. Probably my second favorite would be party-based RPGs, which is another neglected genre these days, last one I really enjoyed there was Neverwinter Nights 2: Mask of the Betrayer. I am not a big fan of things like Skyrim where you control one character and companions are barely an afterthought. One of the things I like most in RPGs is the interactions between you and different characters in your group and among themselves. It adds so much more to the overall story that I enjoy.
But I do have some things to look forward to along those lines right now. Fire Emblem: Awakening comes out this month(Supposedly Monday, but none of the retailers here seem to agree on an actual date) for the 3DS and give me some strategy RPG fun. Later down the road there is Project Eternity from Obsidian, a new game in the style of classics like Baldur's Gate and Planescape: Torment(One of my favorite games of all time). Also Triumph Studios, developer of my favorite turn-based strategy series Age of Wonders, is announcing their next game this week. Triumph has been working on the game since 2009 when they shipped Overlord 2, and I am really hoping its Age of Wonders 3 or some other turn-based strategy game.