What is a p-value?
The p-value is the probability of a particular statistic from the data, or a more extreme one, occurring under the null hypothesis (see e.g. Everitt, 1995). A p-value should not be confused with a type I error which is the probability of falsely rejecting the null hypothesis when it is true. The former does not involve any concept of rejection or acceptance. The type I error on the other hand does. See also the Graduate Statistics talk on power analysis for further discussion.
Reference
Everitt BS (1995) The Cambridge Dictionary of Statistics in the Medical Sciences. Cambridge University Press.