How do I properly interpret a p value?
To interpret the p-value, always start by relating it to the null hypothesis.
One way of thinking about the p-value is that it is the probability of getting the results you are getting, assuming that your null hypothesis is true. If the p-value is very small, this means that the probability of getting the results you get under the null hypothesis is very small. If this probability is small enough, you have evidence that lets you reject the null hypothesis.
For example, if you do a t-test on a sample of 30 tires to see if the air pressure at which tires from the population the sample comes from explodes is greater than 100 psi, and you get a sample mean of 103 psi with a sample standard deviation of 5 psi, your p-value will be .00133. Assume the null hypothesis is true; that is, assume that these tires come from a population with a population mean explosion pressure of 100 psi.
The probability of finding a sample of tires with a sample mean explosion pressure of 103 psi is only .00133-- quite small. If we use the standard rejection area (i.e. alpha) of .05, we can state that the sample we have taken gives us evidence to reject the null hypothesis.
One last thing: always draw a picture when you do this kind of problem.