Null Hypothesis Significance Testing (NHST) is a procedure in which we try to learn something about the data by forming an hypothesis and then ruling out (or "rejecting) that hypothesis. Conceptually, NHST is similar to "deductive reasoning" in philosophy, or "differential diagnosis" in medicine: we can arrive on a single unambiguous conclusion only by ruling out all other possibilities. Therefore we can "prove" that the alternative hypothesis is true by ruling out the only other possibility, the null hypothesis.
In order to perform NHST, have to specify a null hypothesis, typically denoted H0, and an alternative hypothesis, typically denoted HA. H0 and HA must be mutually exclusive for NHST to make sense: if H0 is true then HA must be false, and if HA is true then H0 must be false.
NHST has two possible outcomes: we reject the null hypothesis in favor of the alternative hypothesis, or we fail to reject the null hypothesis. Failing to reject the null hypothesis does not mean that we reject