The Hearthstone alpha was the alpha test phase in the development of Hearthstone. The alpha test presumably ended on August 16th 2013, when Hearthstone began its closed beta testing phase in the Americas region.
An alpha test is the phase of testing before beta testing. It represents a product which is not yet fully functional, but which is being tested among a relatively select group. Once all major features have been implemented and major bugs fixed, the game will likely move on to the beta testing phase, with release to a far larger group of players.
For an exploration of developments and design decisions during the alpha phase, see Design and development of Hearthstone.