Alpha versions of software are normally not released to the public and have known bugs. They are often seen internally as a 'proof of concept'. Avoid alphas unless you are desperate or else being paid as a 'tester'.
Beta versions, sometimes just called 'betas' for short, are a little better. It is common practice nowadays for companies to release public beta versions of software in order to get free, real-world testing and feedback. Betas are very popular and can be downloaded all over the Internet, normally for free. In general you should be wary of beta versions, especially if program stability is important to you. There are exceptions to this rule as well.
 For instance, Google has a history of excellent beta versions which are more stable than most company's releases.