The alpha carbon (Cα) in organic molecules refers to the first carbon atom that attaches to a functional group, such as a carbonyl. The second carbon atom is called the beta carbon (Cβ), and the system continues naming in alphabetical order with Greek letters.
The nomenclature can also be applied to the hydrogen atoms attached to the carbon atoms. A hydrogen atom attached to an alpha carbon atom is called an alpha-hydrogen atom, a hydrogen atom on the beta-carbon atom is a beta hydrogen atom, and so on.
This naming standard may not be in compliance with IUPAC nomenclature, which encourages that carbons be identified by number, not by Greek letter, but it nonetheless remains very popular, in particular because it is useful in identifying the relative location of carbon atoms to other functional groups.
Organic molecules with more than one functional group can be a source of confusion. Generally the functional group responsible for the name or type of the molecule is the 'reference' group for purposes of carbon-atom naming. For example, the molecules nitrostyrene and phenethylamine are very similar; the former can even be reduced into the latter. However, nitrostyrene's α-carbon atom is adjacent to the styrene group; in phenethylamine this same carbon atom is the β-carbon atom, as phenethylamine (being an amine rather than a styrene) counts its atoms from the opposite "end" of the molecule.
Alpha-carbon (α-carbon) is also a term that applies to proteins and amino acids. It is the backbone carbon before the carbonyl carbon. Therefore, reading along the backbone of a typical protein would give a sequence of –[N—Cα—carbonyl C]n– etc. (when reading in the N to C direction). The α-carbon is where the different substituents attach to each different amino acid. That is, the groups hanging off the chain at the α-carbon are what give amino acids their diversity. These groups give the α-carbon its stereogenic properties for every amino acid except for glycine. Therefore, the α-carbon is a stereocenter for every amino acid except glycine. Glycine also does not have a β-carbon, while every other amino acid does.