As an introduction let me run down some of my favorite games and why they are fun to me:
Sim Games: from sim city to the sims, there is a combination of meeting challenges and inventing your own goals
Counter-Strike: fast-paced and team oriented,
Team Fortress: brought class systems to multiplayer
Quake 2 CTF: widely played team first person shooter, with grappling hook!
Unreal Tournament: many choices of weapons, levels, and physical effects
Cyber Nations: persistent browser-based game with thousands of players
Ultima Online: first? mmorpg with players in a live, changing graphical environment
Scorched Earth: the mother of all games
Warcraft/Starcraft: blizzard brings depth to their rts
Command and Conquer: pushing envelope in game storyline, the installer was even part!
Total Annihilation: involving and groundbreaking rts
Half-Life: immerses gamer in the conflict
Final Fantasy: best rpgs ever? characters and worlds that stick with you
Super Mario Brothers: bright, instantly enjoyable response
Legend of Zelda: arcade fun with adventure and thoughtful puzzles
Civilization: requires careful strategy