Because you are posting in the automated testing post, I understand you believe they are automated test methodologies?
Alpha and Beta are "milestone" versions of software and the regression tests that are done on them often
1) are called alpha / Beta tests because of that
2) are executed by the end-users
Alpha versions of software come first, followed by Beta:
Alpha: A very early version of a software product that may not contain all of the features that are planned for the final version. Typically, software goes through two stages of testing before it is considered finished. The first stage, called alpha testing, is often performed only by users within the organization developing the software. The second stage, called beta testing , generally involves a limited number of external users.
Both testing are executed at the final stage of the product. Big difference is design change. In Alpha testing, based on results design may get change at last. But in Beta testing, project design won't change after that. Generally these terms are using in product companies. They have these types of testing in their testing life cycle.
My understanding is that ALPHA testing is carried out by the development company by "releasing" the application internally.
Beta testing is a "release" to external users who hope to gain some sort of edge by have access to new features.
The value of the process is questionable in that there is often no formal fault tracking or reporting and everyone has their own agenda.