How does a hypothesis test work?

A hypothesis test can be thought of as being somewhat similar to a court case. In a court case, we want to find out whether someone is either guilty, or not guilty, so we assume that they are not guilty, and if there is strong enough evidence against them, we deduce that they are guilty. 

In a hypothesis test, we have two mutually exclusive hypotheses (they cannot both happen at the same time) concerning a population parameter - the null hypothesis (we assume this to be true, and try to show otherwise - this is our equivalent of not guilty), and the alternative hypothesis, which gives us more information about our population parameter if it turns out that our null hypothesis is incorrect (our equivalent of guilty). We also decide how strong our evidence needs to be in order for us to be comfortable in saying that the null hypothesis is incorrect (i.e - in declaring someone guilty) - this is called the significance level, and for us, is the probability of incorrectly rejecting the null hypothesis when it is in fact correct (we usually set this to a small probability, such as 0.05). If the probability of our evidence occuring under the null hypothesis is less than the significance level, we deduce that since the probability of this happening by chance under the null hypothesis is so small that something else must be going on, and so we reject the null hypothesis.