Suppose you want to test to see if exam scores between men and women in a chemistry class are different. In that case the test would be two-tailed because we are not specifying a test that men score higher than women or women score higher than men. We are just looking for a difference, which could go in either direction, thus 2-tailed.
A one tailed test would be testing to see if men score higher than women, which would then have a null hypothesis of Ho: Mu1 = Mu2 and Ha: Mu1 > Mu2 where Mu1 is the mens mean and Mu2 is the women's mean. Likewise you could test to see if women score higher than men's score, so Ho would be the same and Ha would be Mu1 < Mu2.
If the significance level of the test is .05, then for a two tailed test, there would be .025 on the left tail and .025 on the right tail, and if it is a Z test, the critical values would be -1.96 and 1.96.
For a right tailed test, still with significance of .05, the critical value for a Z test would be 1.645 (notice it is lower because ALL of the .05 is i one-tail, making easier to reject). Likewise the critical value if this was a left-tailed test would be -1.645.