How is an axiom different from a theorem?

The simple answer is that a theorem is something we "prove", while an axiom is something we accept without proof. In other words, axioms are things we are so utterly convinced of that we don't feel the need to prove them.

Make a list of several things which you accept as "axiomatic". (These do not necessarily have to be related to mathematics.)

Euclid’s Axioms

1. Any two points can be joined by a straight line.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. Parallel postulate. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.