As far as I understand it, the p-value is the probability of obtaining the observed data or more "extreme" values under the null hypotesis, concluding that if this probability is small enough the null hypotesis is likely false.
Why are we interested in the probability of more "extreme" values as opposed to, for instance, the probability of some small interval around the obtained value? (i.e. the probability of obtaining a value in the interval [obtained_value - sigma/2, obtained_value + sigma/2] )
What does "more extreme" actually mean? Sure, if we are talking of a test statistic following a normal distribution, "more extreme" can mean farther away from the mean. However, what about a bimodal distribution? Is there some formal way of stating what "more extreme" means?