The probability of a Type I error is P("Reject "H_0|H_0" is true"). For example, it's when the sample mean is significantly different from 0, when the true population mean is not. P("Type I error") for mu is the chance of the true mu lying outside our confidence interval for it, and this is equal to the area under the probability distribution curve outside the C.I. for mu (e.g. the left and right tails).
The chance of a Type I error occurring is directly related to the width of our C.I. for the parameter. If we want to decrease the chance of Type I error, we increase the width of the C.I., which means decreasing the area we wish to have in the tails, and that is simply done by decreasing the value we use for alpha.
Our alpha-value is actually set to be equal to the total area in the tail(s). Simply put, that means P("Type I error") = alpha. Thus, lowering alpha will mean lowering the chance of Type I error to (100 * alpha)%.
