The alpha and beta of a manager vs. a benchmark are obtained by fitting a straight line to the points in a scatter plot of the market returns vs. the manager's returns. Alpha is the intercept of this straight line, while beta is the slope. Hence, if the market returns change by some amount x, then the manager returns can be expected to change by Beta * x.
Alpha is the mean of the excess return of the manager over beta times benchmark:
mean(i = 1, ... , m)( mi - Beta * bi )
Alpha is a measure of risk (beta)-adjusted return.