A simple regression of asset returns y on market returns x gives y = a + b x + e. The slope b is the beta, the intercept a is the alpha over the period, and e is the residual the model could not explain.
Worked example: regress a stock's 250 daily returns on an index's. If the covariance is 0.000072 and the index variance is 0.000090, then b = 0.80. A 1% index move is associated with a 0.8% stock move on average, which is a statement about averages, not about any particular day.
OLS assumes the errors have constant variance, no autocorrelation, and no correlation with the inputs. Financial data violates all three routinely, which does not make the estimate useless but does make its reported standard errors optimistic.
Related: r-squared, residuals, heteroskedasticity, beta-estimation