Classical Approach
The Classical Approach to hypothesis testing is to compare a test statistic and a critical value. It is best used for distributions which give areas and require you to look up the critical value (like the Student's t distribution) rather than distributions which have you look up a test statistic to find an area (like the normal distribution).
The Classical Approach also has three different decision rules, depending on whether it is a left tail, right tail, or two tail test.
One problem with the Classical Approach is that if a different level of significance is desired, a different critical value must be read from the table.
The PValue Approach, short for Probability Value, approaches hypothesis testing from a different manner. Instead of comparing zscores or tscores as in the classical approach, you're comparing probabilities, or areas.
The level of significance (alpha) is the area in the critical region. That is, the area in the tails to the right or left of the critical values.
The pvalue is the area to the right or left of the test statistic. If it is a two tail test, then look up the probability in one tail and double it.
If the test statistic is in the critical region, then the pvalue will be less than the level of significance. It does not matter whether it is a left tail, right tail, or two tail test. This rule always holds.
Reject the null hypothesis if the pvalue is

You will fail to reject the null hypothesis if the pvalue is greater than or equal to the level of significance.
The pvalue approach is best suited for the normal distribution when doing calculations by hand. However, many statistical packages will give the pvalue but not the critical value. This is because it is easier for a computer or calculator to find the probability than it is to find the critical value.
Another benefit of the pvalue is that the statistician immediately knows at what level the testing becomes significant. That is, a pvalue of 0.06 would be rejected at an 0.10 level of significance, but it would fail to reject at an 0.05 level of significance. Warning: Do not decide on the level of significance after calculating the test statistic and finding the pvalue.
Here are a couple of statements to help you keep the level of significance the probability value straight.
The Level of Significance is predetermined before taking the sample. It does not depend on the sample at all. It is the area in the critical region, that is the area beyond the critical values. It is the probability at which we consider something unusual.
The ProbabilityValue can only be found after taking the sample. It depends on the sample. It
is the area beyond the test statistic. It is the probability of getting the results we obtained if the
null hypothesis is true.