An axiom, sometimes also known as a postulate, is a premise which is accepted as true without proof. It differs from an act of faith in that an axiom is typically either accepted "for the sake of argument", or is simply "self-evident".
A mathematical system of logic seeks to prove statements, typically known as theorems, by combining other statements according to the inference rules defined as part of the logical system. In order to "get started" in this process it is desirable to have statements that don't need to be proved. These are the axioms. (It is possible for a system of logic to have no axioms, but most do.)
For a specific example, suppose we want to devise a system of logic to prove statements about arithmetic, we might define rules of inference like
- "if a is an integer, then a+1 is also an integer", and
- "if a and b are both integers, and a = b, then a+1 = b+1".
But this does not get us very far unless we also define some axioms as starting points — namely
- 0 is an integer, and
- 0 = 0.
By combining these axioms with the rule of inference, we can prove theorems like
- 0+1 = 0+1,
- 0+1+1 = 0+1+1,
- 0+1+1 is an integer,
and so forth.
Note that an axiom need not be true in any obvious sense. Rather, axioms are accepted for the sake of argument. One famous example of this is Euclid's parallel postulate. Many mathematicians for centuries thought that it was a theorem that could be proved from the other axioms and definitions Euclid used as the basis of his system of geometry. All attempts failed, however, and eventually it was discovered that the statement need not even be true in the first place. This realization lead to so-called "non-Euclidean" geometries in which the axiom is not used.
Therefore, if a mathematician demonstrates a theorem in a given system of logic, what he or she is really saying is, "if these axioms are true, if you agree with these definitions, and if these rules of inference are valid, then this theorem must necessarily be true as well." To the extent that the axioms and rules of inference are "true" in real life, and to the extent that the definitions describe "real" things, the theorems allow us to make deductions that will also be true in the real world.
In philosophy, axioms are the fundamental assumptions upon which a philosophical system or set of arguments is based. For example, a system of ethics might be based on the axiom "everyone has a good idea of what other people like and dislike."