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There are a few stages in product development and alpha is one of these development stages.
A product in alpha stage will satisfy all software requirements but will not have been completely tested or it could possibly be missing minor features.
Products in alpha are considered to be in the first major stage of the so-called "release cycle" and that is where the name "alpha" comes from. Alpha is the first letter of the Greek alphabet.
Products in alpha are generally considered to be only 35-40% complete and will be very buggy. Alpha releases are generally not released to the public but instead to software testers or a community of dedicated past users.