Move α is a feature of the so-called Revised Extended Standard Theory (REST) of transformational grammar developed by Noam Chomsky in the late 1970s. The term refers to the relation between an indexed constituent and its trace t, e.g., the relation of whom and [t] in the example
In the 1990s Minimalist Program, it became a structure-building operation together with "Merge". "α" is the placeholder symbol for the moved constituent. The constituent (whom) and its trace (t) are said to form a "chain".
"Move α" in practice means "Move any constituent anywhere". "Affect Alpha" is a generalization to the effect of "Do anything to anything". The latter is viewed with suspicion by proponents of REST as an overgeneralization.
"Move α" marks a shift of attention in transformational grammar in around 1980, away from focussing on specific rules (the only "rule" is "Move α") to "Principles" constraining them.