An axiom P is independent if there is no other axiom Q such that Q implies P.
In many cases independency is desired, either to reach the conclusion of a reduced set of axioms, or to be able to replace an independent axiom to create a more concise system (for example, the parallel postulate is independent of Euclid's Axioms, and can provide interesting results when a negated or manipulated form of the postulate is put into its place).
Proving independence is usually a simple logical task. If we are trying to prove an axiom Q independent, then the set of all the other axioms P can't imply Q. One way of doing this is by proving that the negation of the set of axioms P implies Q, it then follow by the law of contradiction that P can't imply Q, because if that were the case then P and not P would both imply Q, and that would be a logical contradiction.