Making a reasonable guess based on some evidence (such as particular examples) is important, but in the end, conjectures need to be justified. This means reasoning on the basis of facts already established in order to convince yourself and others of the validity of the argument. The aim is to show that no matter how some things change, some properties remain invariant.
There are named types of proof, such as proof by induction or proof by contradiction, but more generally, when mathematicians talk about proving, they are referring to the process of justifying a statement in such a way that we can be certain that it is true.
