In independent games, or just "Indie" Games, The alpha stage is a low cost release to gain popularity before the game is released as a retail copy. Players go into testing understanding that the game is still in testing, and not a finished release. It also allows Indie game companies to receive bug reports from potentially thousands of testers, that they do not have to pay to do so. In fact, many indie game development companies charge for the Alpha copy.
In normal development, (Your more famous, and big game development companies) Alpha is the phase where paid employees test and release early videos and screenshots for marketing and bug fixing purposes. Beta is the final phase of testing, this is where the game has been more defined as far as gameplay and graphics go. It allows the company to continue to do bug fixes and patches to fix such bugs, while focusing energy on better defining, graphics, sound, and story lines. It also allows them to work on additional gameplay fixes and bells and whistles.
The Patch numbers are different from game to game. It depends on how the outline for the game has been pitched. The numbers are more or less meaningless. They are just bookmarks for where and when something happened in testing. Generally there will be an Alpha 0.1 - 3.0, Beta 0.1 - 3.0 Retail release usually starts at 1.0.