These are the games that I play routinely

Diablo 2
http://www.blizzard.com/diablo2
An older, yet still exciting, game. Diablo 2 is a game where you control your hero and complete a series of missions to beat the game. There are 3 difficulty levels: Normal, Nightmare, and Hell.
Tyrants Anarchy
http://www.tyrantsanarchy.co.uk
A fun browser based game. All players have a chance to be the best without much effort at all. This is fast paced and not a clicking game.
Wesnoth
http://www.wesnoth.org
A computer based game that is much like Final Fantasy. You control a 'hero' who buys units that then go out to attack your enemies. You control the pace of this game by how many men you have (and what levels they are) in relation to your opponents.
Command and Conquer
http://www.eagames.com
A series of computer games that started with you harvesting tiberium and now control nuclear warheads. Its fun, captivating, and you can get caught up for hours in this game.
Lord of the Rings Trilogy
http://www.eagames.com
A computer game series. Based on Lord of the Rings trilogy by J.R.R. Tolkien it is set in Middle Earth and you follow the adventures of the Company of the Ring.
Astro Empires
http://beta.astroempires.com/?ref=B.18098
This is an online space faring game. You build buildings and bases to increase your fleet, technology, and economy. I am in the [Zerg] alliance (Guild link). The game is fun, fast paced, and fairly easy for anyone to understand.