Alpha is the first phase of extensive testing (there's also some spot testing of specific code/functions prior to and also during, known as Unit or Smoke Testing, depending on who's doing it.) There's some back and forth on the definitions of Alpha testing, generally it's developers or experienced technical testers looking at the functions of the code rather than all the quest and story detail stuff. The definition from a purely I/T perspective has been somewhat bent due to development houses pulling in gamer volunteers to help in Alpha. For those who haven't done it before it can be a little disconcerting entering a building and finding it to be all "wire frame" without any texture. Not uncommon in that phase.
Beta is when the code and a decent percentage of the game experience has been tested, there are still known defects that are being worked (sometimes a lot of them), but should be aesthetically complete. Caveat with those is that it's still very possible to come across game breaking bugs.
The "final," which again in today's world takes on a definition that seems to be set by each Development House is sometimes called a Production Release or Release Candidate. There was a time, when a release candidate wasn't actually put out for public consumption until it went through another phase of testing. New development methodologies (which I don't agree with) push everything out faster, basically using the consumer as the final tester.