In order to explain the importance of Half-Life, I'd like to take you back in time to Doom. Witness the formula of Doom: You begin a level. You have a pistol. There are monsters. As you progress through the level you might find more weapons. You might find some locked doors and some keys to open those doors. Eventually, you hope to find the switch at the end of the level. Flip it, and the level ends. Isn't that a nice screen of statistics?
Repeat. Many, many times. As you get farther in the game, the enemies will get larger and more numerous, and your weapons will get bigger, too. This is the entirety of Doom; in fact, it's also the formula used in the earlier Wolfenstein 3D; in fact, it's a formula that goes back to 2D gaming.
Jump forward a few years, to Quake. Now take the formula of Doom and add a much more advanced 3D rendering engine, capable of such amazing feats as polygonal monsters and rooms on top of other rooms. Nothing like it has ever been seen before1. Also add far more advanced Internet multiplayer options. The fact that the basic formula of Doom hasn't changed one iota is barely noticed, let alone questioned.
Jump forward a few years and a few sequels again. Those multiplayer options have turned out to be the driving force in the computer gaming industry, and the existing major FPS series are moving more and more in the direction of focusing on the multiplayer content exclusive of all else. By estimation of most at this time, certain new multiplayer-only titles would become the meterstick by which all other games would be measured.
Then Half-Life came out and dominated the market. Quake III Arena and Unreal Tournament were successful, but they would never hold the top spot.
It did so by offering a single-player campaign that told a coherent story. When you start the game, you're not thrust into some moon-base with a pistol in hand, evil demons lurking around the corner; no! You are on your way to work. You pass your co-workers, a security guard offers to buy you a beer, and you go about your day. Only then does hell break loose, and even then you still have an objective (get to the surface, get to the Lambda Complex, etc). Scripted sequences helped, too: never is the player removed from Gordon's head, and only once (with excellent reasons) is control removed from the player's hands. Perhaps the biggest thing Half-Life did was to introduce very nearly seamless level transitions. Gone are the post-level statistic screens of the Doom formula. Instead, Gordon just keeps moving through the Black Mesa complex. And all of this is simply done in the confines of a first-person shooter. There are no subscreens, no inventory, no objectives screen. Just a man with a PhD and a gun.
[I've received some /msgs to the effect of "There are other story-driven FPSes that pre-date Half-Life, like System Shock," so I think this last point bears repeating: Half-Life is just an FPS and nothing else. It has no cutscenes, no text to read, no in-game emails to peruse, and yet it still does everything it does. This was its accomplishment.]
The effect of Half-Life on the first-person shooter has been approximately as big as when movie directors first realised they could move the camera around. Nearly every FPS since Half-Life has adopted some of its features or has attempted to improve on them.
If this weren't enough, Valve went and added support for user-made mods that would be the envy of all the other offerings of the time. Quake's classic mod Team Fortress would see a remake in the Half-Life engine that would quickly supplant the game's mindlessly simple built-in deathmatch mode. Valve would later take TFC into itself, and the game would later be included in a patch. Later, a little user-made mod called Counter-Strike would be the single most popular action game on the Internet for a period of years. (A new version of Counter-Strike would later become Half-Life 2's built-in default multiplayer mode.) The effect of these mods was to prolong the game's popularity until the sequel came out six years later, a timespan utterly unheard of in the gaming world.
As I write these words, Half-Life 2 is fresh on the scene, and the days of Half-Life's glory are something of a memory. Most of the major Half-Life mods have announced plans to move to the Source engine, and the popularity of Half-Life online has waned with the rise of so many newer games. But Valve has every reason to hold on to the top spot. Hail to the king.
1 Note that this is not strictly true: many of Quake's features were seen in previous games (Descent leaps to mind), but no previous FPS had packaged all of its major features into one package as it did, and none had Quake's mod support.