Going to wander into obscure territory here. (taking longer than I thought to locate it)
There already exists a good summary and 'researched' description here. I generally agree with it, but wish to add some material.
Half-life is the standard method of describing the rate of radioactive decay of a specific isotope. It is also the name of the very popular first-person "shooter" game that effectively put Valve Software on the map. Two other software developers made "Equels" - my own word for a "sequel" - that occur in the same place and time as the first one. Half-Life featured a nuclear physicyst named Gordon Freeman. Opposing Force featured Corporal Adrian Sheppard. Blue Shift featured Black Mesa Security Officer Calhoun. When playing the "Equels", you do not actually meet Gordon Freeman, but the developers allow you very brief glimpses of him - if you are paying attention.
Most recently, Half-Life has been succeeded by a proper sequel in Half-Life 2, and the subsequent "Episode 1" and "Episode 2". Valve has stated that offering the follow-on adventures in a more modular installment allows for more regular updates to the saga. I expect that it also allows them to put more cutting-edge technology into the latest installment, rather than delaying a big project by re-writing the underlying code. As far as I am concerned, whatever they want to do is fine by me.
Half-Life features Gordon Freeman grappling with the aftermath of a bizarre laboratory mishap. Apparently a portal to an alien world was opened releasing dangerous creatures into the Black Mesa complex. But was it an accident? What really happened at Black Mesa, and who is the mysterious "G-Man" that keeps surfacing in the background, always seeming to have more control over the situation than Gordon does? We get the idea that a powerful being called "Nihilanth" became aware of this world and used the energy released inside the test chamber to open a portal to Black Mesa. Yet that fails to explain the identity of the G-Man.
At the end of Half-Life, Freeman comes face to face with the G-Man, who offers praise, but no explanations. He does, however, offer Gordon Freeman (you) a job. If you accept, you are left in a bizarre place, and apparently you perish if you refuse. At the begining of HL2, the G-Man appears and speaks to you before you find yourself on a train, about to arive in City 17. Has the G-Man kept Gordon in stasis all this time? And odder still is that the Vortigonts who we remember as nasty enemies in HL, are now co-conspirators, and actually treat you with a reverence approaching adoration. What the hell has happened?
The Vorts are grateful for their liberation from bondage to the Nihilanth. But there is more, much more. In HL2, talk to a Vort after key dialogue has finished and its time for you to move on. I recommend enabling subtitles for this, and talking to the Vort that just installed the gun on your airboat. There are other opportunities, but theres just too much background noise at New Little Odessa, and the Vort there has to wait for breaks in Colonel Cubbage's endless stream of radio hails to speak. The Vort will speak in prose and allegory, and make the clearest references to the Vortessence before Episode One. The first thing that the Vort will say is: "We remember the Freeman: we are coterminus."
Coterminous: sharing the same boundaries, having the same span or scope. What an odd thing to say.
Gordon Freeman (you), of course says nothing, the other characters talk to, or near him, usually presenting expostion to explain your situation, or its background. The enigmatic part is that you are put in a number of places with no clear objective save what the minor character tell you. [Might as well call them 'meatlings' as most of them die anyway.] Since you do not speak, and in many cases you (as the player) have no forenowledge of what to do in some situations, there is a vaguary of identity. Is this really Gordon Freeman, or someone else? Someone that might truly not know what they are doing?
This is never more clear than in the very begining, at Black Mesa. You are apparently late for the test procedure, but you have no clear idea where you should be going. Fortunately, people that you meet tell you brusquely where you should be, and there are color-coded lines on the wall that you can follow to your destination. When you arrive, the scientists speak to you in a slightly formal manner, not exactly as if he has done this hundreds of times before. Then at the crucial moment, you can see what needs to happen, and where the crucial object needs to go. I remember thinking - "Can this be right? I just do it this way?"
Of course, when I do perform the task, everything goes to hell. Did I do it wrong? Did I cause this catastrophe? There turns out to be other reasons why it happened. Yet that does not mean that what I was doing, as Gordon Freeman, was correct.
At the very begining, before anything happens, we are told that the identity of Gordon Freeman's Administrative Sponsor is classified, and that his disaster response priority is discretionary. Discretionary? Isnt he just a theoretical physicist? Surely we were informed of these vaguaries for a reason?
I think Gordon Freeman may not be a physisict at all, but an agent of some kind. His (late!) insertion in Black Mesa was a trial run of some sort, which worked well. At least the G-Man was pleased enough to preserve him for later use. The G-Man, as we see in Episode Two, seems to have ideas that diverge from those of the Vortigonts, and I think this is why the beginning of HL2 played out the way it did. Had Barney not interfered, Freeman would have gotten onto the express train to Nova Prospekt. This is not an accident, it was the only train there, and the player has no real option but to go that way. Then, when finally in Dr. Kleiner's laborotory, its true that "Hedy Lamarr" the headcrab interferes with the procedure. One of the scientists however also says "Something is drawing him away!" and Gordon Freeman appears in Dr. Breen's office not just one time, but twice, before ultimately ending up back at Dr. Kliener's laboratory. In retrospect, being teleported twice to Dr. Breen's office hardly sounds like a random failure.
When Freeman is fighting [your] way through Nova Prospekt, (he got there anyway, but by way of the Resistance) Dr. Breen can be heard over the public address system. He discusses the state of affairs regarding Overwatch's capture of Dr. Vance, but their failure to get Dr. Freeman. [Its a Utopian Dystopia where the world seems to be in decay, but the most prominent figures are physicists.] Dr. Breen says that Dr. Freeman's condition; between the episode at Black Mesa and his current efforts against the Combine, precluded the development of any combat abilities. In short, Dr. Breen knows that the G-Man has been keeping Freeman in stasis. Given the train schedule, the teleport 'fiasco', and Dr. Breen's unexplained knowledge of Freeman's situation, I believe that the G-Man was intentionally sending Freeman to Dr. Breen. Possibly at the request of the Combine Advisors.
What precisely is the meaning of "Half-Life"? Does it infer that time is of the essence? That things are decaying and falling to ruin all around him? That Freeman has a short life expactancy? Or perhaps it means that Gordon Freeman is only half human. Half living, half not living, possibly a prototype Combine soldier. Now a soldier without orders, who can only react to his situation.
Or no life at all.