Alpha (α) is the distance of each axial point (also called star point) from the center in a central composite design.
Alpha, along with the number of center points, determines whether a design can be orthogonally blocked and is rotatable. Orthogonally blocked designs let model terms and block effects be estimated independently and minimize the variation in the regression coefficients. Rotatable designs provide the preferred property of constant prediction variance at all points that are equidistant from the design center, thus improving the quality of the prediction.
Minitab's default designs achieve rotatability and orthogonal blocking when both properties can be achieved at the same time. When the design includes blocks and you cannot achieve both properties at the same time, the default designs use α such that the design includes orthogonal blocks. When there are no blocks, the default designs use α such that the design is rotatable.