An axiom is a particular kind of sentence, that forms the basis of a logic. Logic describes any method for establishing true conclusions from true premises, and is broadly subdivided into deductive logic, which draws specific conclusions from specific premises, and inductive logic, which draws general conclusions from general premises. The truth of the conclusions in any logic is determined by the soundness and validity of the steps connecting them to the initial premises. The overall logical process by which conclusions are induced or deduced from premises is an argument; ultimately, every logical argument begins with one or more axioms.
The thing that makes an axiom so special is that it can't be proven by any sort of logical argument. You just have to accept an axiom is true, and go from there. For this reason, axioms are generally statements that seem self-evidently true; statements we find it hard to imagine could be false. The logic most people understand and recognize as logic was formalized by the Greek philosopher Aristotle. His logic rests on a small number of axioms that state seemingly uncontrovertible assertions: the Axiom of Identity states that a given A is the same as itself, so A = A; the Law of Excluded Middle, which is actually derived from the identity axiom and so not strictly axiomatic itself, asserts that a given A must either be the same as a given B, or different from it, so A = B or A = ~B (~B means 'not B' in formal logical notation). Most of Western philosophy accepts these axiomatic principles as Aristotle formed them; but it's worth remembering that there is no logical reason that we have to. We can, for example, emulate Oscar Wilde and assert "nothing is itself alone" (it should be clear that this denies the validity of the identity axiom); we can posit that there may be a class of things which can both be and not-be one another simultaneously. It may seem absurd, to modern people conditioned to accept the assumptions - the axioms - of empirical scientific method, but it is entirely valid; even, for some purposes, advantageous.
Axioms are enormously powerful, because when we choose an axiom we are creating a whole potential logic that rests upon that axiom. It is important to distinguish between an axiom, which is unprovable within a logic, and an assumption, which is a statement whose truth is tested by logical analysis. For example, "cats have antlers" doesn't work as an axiom, since we can inductively show that there exists no cat in particular which has antlers, and so conclude that cats in general share this lack. It is a false assumption. "Cats do not have antlers" is not axiomatic, either, although it is so obvious empirically to anybody who understands what cats and antlers are that it can be treated as such under most circumstances. The danger lies in treating what seem to be obviously true assumptions as axioms - this is a good way to close ourselves off from valid alternative hypotheses. Ironically, this sort of confusion is common among adherents of the scientific method - which properly adopts a spirit of open enquiry, continuously challenging its own assumptions.
In common parlance, an argument is a dispute, a difference of opinion; each side has a reason for believing its conclusions to be true, and each side believes the other's conclusions to be false (this can be seen to be a function of Aristotle's Excluded Middle - without that, each side could have an opinion that worked for them, and not invalidate the other's opinion thereby). Some of the most intractable arguments concern situations that resist logical analysis - theology is a fertile ground for these, beginning with the fundamental argument over whether God exists. The atheist position that there is no God is grounded in an empirical worldview, which asserts that everything we observe can be explained without recourse to a God. A rule called Occam's Razor advises us that if we don't need an extra element to account for something, we should properly discard it, so that our explanation is as simple and elegant as possible. This rule, suggesting the simplest possible explanation for a phenomenon is also the best, has disadvantages which may become clear on a little reflection; we can readily conceive of situations that arise from complicated interactions but are explicable by more simple ones, and this is for example one facet of the magician's art. These performers rely on the assumptions people make based on what they observe; they are skilled at manipulating those observations, misdirecting their audience, and producing apparently 'magical' results that are explicable only by recourse to hidden phenomena of the sort Occam disregards. It should be said that the simplest explanation - "it's magic!" - is, from an atheist's perspective, closely similar to the theistic explanation - "God makes it happen." Scientific atheists assert that what we cannot explain with our current level of knowledge about the universe will become explicable as that knowledge increases and we appreciate the greater complexities of time and space. There are reasons to doubt this, and certainly no logical reason why we must accept it, but it is as logically valid a position as its counter from the theistic perspective.
Even among theists, arguments are rife, precisely because the dogma of each religion is beyond the scope of logical refutation. Where logic derives its authority from its axioms, a religion derives its authority from the Divine - belief in its deities is therefore a necessary precursor to acceptance of the religion's claims, although in this religion functions exactly as logic does: for we will only find logical force to an argument if we first accept the axioms that underpin it. Major religions differ in their characterizations of the Divine - some are monotheistic, believing in just one God (for example, the Abrahamic God worshipped by Christians, Jews, and Muslims); others are polytheistic, believing in a whole pantheon of Gods (Norse Pagans, Hindus, and Buddhists are examples). Within a given religion, there may be differences on small points of dogma, which result in schisms or splits. These can be extremely violent. A schism within the Catholic Church of medieval Europe led to the establishment of a rival Papacy in Avignon - these Popes were known as Antipopes because they stood against the temporal authority of the Popes in the Vatican, who were officially the earthly representatives of Christ. Individuals who challenge the dogma of a religion, as Martin Luther did with his 95 Theses, or as Galileo did with his assertion "eppur si muove", can find themselves accused of heresy - under some theocratic jurisdictions, this can be a crime punishable by death. The Albigensian Heresy of the Cathars, knights who founded the first central bank in history, led to the destruction of that immensely powerful order (although several modern secret societies, like the Prieure de Sion, claim descent from them).
It comes down to this: everything any of us believe, no matter what authority we claim for it, is simply a matter of opinion. And we all know what opinions are like - something else that begins with A...