Portal (Multiplatform, 2007)

Valve is a company that is always pushing the limits of game design.  Their Half Life series raised the bar for first person shooters, and video game storytelling as a whole.  Team Fortress 2 set a new standard for online multiplayer games.  It is funny too, that their most influential game was not even intended for its own individual release.  Portal (2007) was first released as part of The Orange Box, a compilation of Valve games which also included every part of Half Life 2 and Team Fortress 2.  Portal was developed by a tiny faction of Valve's staff.  It was intended to be just a bonus feature.  What it became was a revolution.
You awake from a hypersleep chamber into a tiny glass containment unit, with only a toilet, cube, and radio inside.  An electronically enhanced female voice talks to you over a loudspeaker.  She is GLaDOS (Genetic Lifeform and Disc Operating System), and she is going to have you perform some experiments.  After short circuiting briefly, she creates a portal on the wall inside your cell and on a wall outside the cell.  Before you step through, you look inside and see yourself standing inside of the cell.  This portal is not a door, it is a wormhole.
This is the play mechanic for the whole game.  Using portals, you must solve a number of puzzles to reach the game's conclusion (where you are promised cake and brief counselling).  To avoid this becoming too disorienting or confusing, the game is expertly designed.  The first time you have to use a portal, the designers make sure that you see yourself, so you understand that you are not entering a new space.  When you go into one portal you come out of the other.  You slowly become acclimated with each test until you are finally given the portal gun (pictured above) and have control over where you place the portals.
So this is essentially a puzzle game in first person view, but it is more than both of those things.  Portal imagines a world run by Aperture Science, a testing facility that uses humans as its subjects, and you are possibly the last human.  Your only contact is the increasingly sadistic and sarcastic GLaDOS, who is both hilarious to listen to and threatening at the same time.  She is in charge of the entire facility, but her motivation is not made entirely clear (In Portal 2, it is explained that whenever you solve a test, she gets a euphoric stimulant).  Even though you don't ever see her until the very end of the game, she is one of the most memorable video game characters and villains.
Solving tests is a rewarding challenge too.  There are so many possibilities for what you can do with a portal gun, and Valve takes advantage of all of them.  Need to put a weighted cube on a button that is out of reach?  Put a portal above the button and another on the floor, and drop the cube into it.  Need to get somewhere that is too high to jump to?  Put a portal far below you and another on a different surface, jump into the lower one, and your momentum will fling you up to the high platform.  As you progress, tests become more intricate, and require more precision and masterful timing.
I'm fairly sure that the central theme of this game is that brains beats brawn.  The first person genre is famous for being entirely centered around combat, but Portal features none at all.  There are some things that will try to kill you, like the adorable turrets (passive aggressive, and speaking in a high pitch bubbly voice), but you never truly fight back.  You always have to beat foes by being smarter, quicker, and using portals.  Even in the final battle with GLaDOS, you don't physically attack her.  Using portals, you make her rockets change direction and hit her instead.  She likes to think she is more intelligent than a human, but she has belittled herself at this point to brutality and empty insults ("You're not smart!  You're not a doctor!  You're not even a full-time employee!")  Perhaps more modern games will take a page from Portal and foster intelligent design and problem solving, rather than gun-packing violence.