What is the meaning of p values and t values in statistical tests?

After taking a statistics course and then trying to help fellow students, I noticed one subject that inspires much head-desk banging is interpreting the results of statistical hypothesis tests. It seems that students easily learn how to perform the calculations required by a given test but get hung up on interpreting the results. Many computerized tools report test results in terms of "p values" or "t values". How would you explain the following points to college students taking their first course in statistics: What does a "p-value" mean in relation to the hypothesis being tested? Are there cases when one should be looking for a high p-value or a low p-value? What is the relationship between a p-value and a t-value?