An axiom is a proposition that is assumed to be true, because you believe it is somehow reasonable.

Here are some examples:

Axioms 1. if a = b and b = c, then a = c

This seems very reasonable! But, of course, there is room for disagreement about what consitutes a reasonable axiom. For example, one of Euclid's axioms for geometry is equivalent to the following:

Axioms 2 (Parallel Postulate) Given a line l and a point p not on l, there is exactly one line through p parallel to l.

In the 1800’s several mathematicians realized that the Parallel Postulate could be replaced with a couple alternatives. This axiom leads to “spherical geometry”:

Axiom 3. Given a line l and a point p not on l, there is no line through p parallel to l.

And this axiom generates "hyperbolic geometry".

Axiom 4.  Given a line l and a point p not on l, there are infinitely many lines through p parallel to l.

Arguably, no one of these axioms is really better than the other two. Of course, a different choice of axioms makes different propositions true. And axioms should not be chosen carelessly. In particular, there are two basic properties that one wants in a set of axioms: they should be consistent and complete.

A set of axioms is consistent if no proposition can be proved both true and false. This is an absolute must. One would not want to spend years proving a proposition true only to have it proved false the next day! Proofs would become meaningless if axioms were inconsistent.

A set of axioms is complete if every proposition can be proved or disproved. Completeness is very desirable; we would like to believe that any proposition could be proved or disproved with sufficient work and insight.

Surprisingly, making a complete, consistent set of axioms is not easy. Bertrand Russell and Alfred Whitehead tried during their entire careers to find such axioms for basic arithmetic and failed. Then Kurt Godel proved that no finite set of axioms for arithmetic can be both consistent and complete! This means that any set of consistent axioms is necessarily incomplete; there will be true statements that can not be proved. For example, it might be that Goldbach’s conjecture is true, but there is no proof!