Then there are the hardcore games. Eve Online, World of Warcraft, Call of Duty, Grand Theft Auto, Street Fighter 4, etc. These games often require lots of skill or dedication, and are often big budget titles.
But there is a ton of stuff in between. Games that you might be able to play fast but are very difficult (like Spelunky), that the player can choose how to play (Minecraft or Terraria), that just aren't designed to be highly competitive and difficult (Sly Cooper series, maybe Ratchet and Clank series), aren't hardcore ultra-deep sims (SimCity, Game Dev Story), and many sports games (Madden, NBA2k, etc).
Some people go really deep on some games and play them in a 'hardcore' way. The lines aren't easy as I know many people think World of Warcraft has been dumbed-down throughout the expansions to where it may not be 'hardcore' for many people.
But that's my general taxonomy. Casual, 'normal', hardcore.