Here is the wikipedia definition of both:
Alpha Code – Alpha is the stage when key gameplay functionality is implemented, and assets are partially finished. A game in alpha is “feature complete”, that means the game is playable and contains all the major features the game will have upon release. These features may be further revised based on testing and feedback. Additional smaller, new features may be added. Similarly planned, but unimplemented features may be dropped at the Alpha Stage. Programmers focus mainly on finishing the codebase, rather than implementing additions. Alpha occurs eight to ten months before “Code Release” (where the game in it’s final shape, in Master Disc form, ships off to the manufacturer to be produced and copied into the discs you see on store shelves).
Beta Code – Beta is a feature and asset complete version of the game, when only bugs are being fixed. This version contains no bugs that prevent the game from being shippable. Generally, no changes are made to the game’s features, assets, or code. Beta occurs two to three months before “Code Release” (described above).
So, the video you see above is from Half-Life in it’s Alpha Version, released as part of a technical demo of the game to show interested publishers, videogame media, and the like. That way they knew what they could expect from this new Valve-developed FPS strangely named “Half-Life” (whatever that meant). Before it changed the world, of course. And “Steam” was just a lofty dream in a young Gabe Newell’s brainy noggin. Pretty cool eh!