Let's see the classics...
-Half Life: You take the role of Dr. Gordon "Earnest" Rutheford in a game where the US captured an intact UFO, and your job is to figure out how it works before the extremely unstable isotope powering it deteriorates into lead. (Puzzle game with a ticking clock)
-StarCraft: It is a simulation game where you are the lead designer of the Magrathean luxury-planet industry and you have to make specialized luxury-planets for your customers. Dolphins are optional. (Sandbox worldbuilder + management sim)
-Tomb Raider: You lead a clan of rugged tomb raiders in Egypt from the 19th century till today, starting out as a small band of desperate men and aiming to become a successful and rich underground smuggling ring. (Management sim)
-Street Fighter: Set in a civil war torn nation, your goal is to capture the capital by using tactics and management to take over the city from your enemies in urban warfare, one street at a time. (Turn-based tactical RPG)
-Mortal Combat: The gods of various pantheons want to duke it out to decide who should rule the entire world, but they are afraid that doing so by themselves would destroy the world they are trying to rule, so they chose mortal champions to do the deed. You are one of these chosen champions, and your goal is to secure victory for your god while getting stronger and also pursuing your own goals. (JRPG)
-Max Payne: A new side branch in the Phoenix Wright franchise, you are to take the role of the young and idealistic cousin of Winston Payne. He grew up as a fan of Phoenix, but still became a prosecutor. During the events of the game, he would try to find the truth instead of going for a guilty verdict alone and would try to work with the defense attorney to reach this goal, but his efforts are hampered by the deep-running antagonism between the two sides. In the game's last case, he would have to go up against Phoenix himself (after his return to the court after Apollo Justice) and the two of them must find a way to do what Phoenix does best (acquit the defendant while still finding the culprit) under pressing time-constraints. (Puzze visual novel)
And now, for some more recent ones:
-Half Life 2: In the first game, you figured out how the alien ship works. Now, it turns out that it was only a scout, and the mothership is coming. Your job is to use the captured alien technology to devise a device that could be used to emit radiation that would square the half-life rate of the alien power source, rendering it useless, before they arrive and take over the planet. (Puzzle game with a ticking clock)
-Mass Effect: A first person non-linear puzzle game in the vein of Portal, where you have to escape from an abandoned industrial complex using physics and your own wits. Snarky female commentator is optional. (First person puzzle)
-Dragon Age: You are a young dragon who just came into age to have your own lair. Manage your lair! Hoard your wealth to attract the dragon ladies! Torch villages and fight knights for experience and loot, or help the villagers in need and the knights during war, and they might just start honoring you with gifts and start to worship you! The choice is yours! (Sandbox game with optional raising-sim and dating-sim elements)