Borrowed from traditional (i.e. non-games) software development, Alpha designates a key stage in a game's development. The definition of Alpha for traditional software is, according to Wikipedia:
Alpha software can be unstable and could cause crashes or data loss. [...] The alpha phase usually ends with a feature freeze, indicating that no more features will be added to the software. At this time, the software is said to be feature complete.
In game development, the definition is a bit more blurry.
Yes, the exact definition of an Alpha may (and often will) vary from one developer to another, and also from one publisher to another. The Alpha build being a key deliverable when you've got a publisher deal, you better be crystal clear with the publisher about what needs to be implemented in that build. This will avoid any surprises.
A game's Alpha build usually complies with the following points:
- The game is playable from start to finish.
- All the game's features are implemented to at least first pass quality.
- Some art may still be place holder.
- Crashing bugs may be present.
The important thing after is that after the Alpha build has been approved, the developers should stop adding features. However that's theory, and as projects tend to be late, Alphas are usually not what they should be. As such, feature implementation might still happen after the delivery of the Alpha.
Also, some features might be added after Alpha simply because the game needs them to not suck (though in that case, people should have realized it sooner).
Although you need to be able to reach the end of the game, it doesn't mean the entire content needs to to be in the build (i.e. side quests, collectibles, etc.). But then again it all depends on what you've defined with your publisher in the contract (or the goals you set yourself if you're an indie developer).
For that reason I don't really like using the word Alpha. Microsoft Games Studios prefers using a term we find in the Wikipedia definition, and which is (to me) much more explicit: Feature Complete.