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.