How does the choice between a one-tailed and two-tailed test change the p-value for a t-statistic?

A two-tailed test looks for results at least as extreme in either direction. A one-tailed test looks only in the direction stated by the alternative hypothesis.

For example, if a positive t-statistic is evidence for the stated alternative, the one-tailed p-value is usually about half the two-tailed p-value. If the alternative is in the opposite direction, the one-tailed p-value will be large because the statistic points the wrong way.

The key CFA habit is to read the alternative before interpreting the p-value:

H1: parameter != 0  -> two-tailed
H1: parameter > 0   -> right-tailed
H1: parameter < 0   -> left-tailed

Do not reject just because the absolute statistic looks big. Compare the correct p-value with alpha.