What is an axiom in layman's terms?

In mathematics, every result known descends from something else: it is proven to be true from other facts.

The one exception is axioms: these things we choose to accept without proving them.

We have to choose some axioms, since we cannot prove anything with nothing, but we try and make them as simple and obvious as possible.

For example, Euclidean geometry rests on five axioms, the first of which is "given two points on a plane, it is always possible to construct a straight line passing through these two points". Another states that it is possible to draw a circle with any center and radius.

Using these simple statements, Euclid then proceeds to prove more complex properties of figures on the plane.

In former times, axioms were considered to be statements that are so simple and "obviously true" that they cannot be proved (or any attempt to prove them would need to be based on more complicated things - and why bother proving it at all if it is obviously true?).

In today's understanding, an axiom is a statement that is, for the sake of developing a specific theory, taken for granted. For example, the axioms of Euklidean geometry (there is exactly one line passing through two given distinct points" and so on) can be used to rigorously prove all theorems of Euklidean geometry. There is but no inherent "truth" to the theory as such. However, if we verify that the axioms of Euklidean geometry hold for some things or phenomena (e.g., for points and lines and drawn on the Earth's surface) then we can be sure that also all theorems (such as Pythagoras) apply to these. But as soon as we notice that "straight lines" on the Earth surface such as meridians are not as parallel as they should be, the Euklidean theory is no longer fully applicable to these phenomena.

Nevertheless, a "good" axiom system is one that allows us to apply it to interesting things. As among those interesting things are the theory of natural numbers, geometry, and set theory, certain axiom systems (Peano, Euklid, Zermelo-Frenkel) relating to these theories have become somewhat standard.

To a mathematician, or a philosopher, in the pre-modern era (and to some crackpots of the modern era) an axiom refers to a truth so basic that it cannot be proved or argued about from other truths. A basic elementary fact to 'obviously' true that the only way to argue about it is to tautologically proclaim "it simply is true!!!". For instance, the fact that addition of natural numbers is commutative, namely that $m+n=n+m$ for all natural numbers $m,n$, might be taken as an example of an axiom in that sense about the natural numbers.

In the modern era, we are no longer obsessed about what things really are (whatever that means) and thus we no longer treat axioms as described above. Instead, we recognise that mathematics is a formal system consisting of a language in which to express yourself, a logical system one uses to manipulate statements, and an initial collection of statements we take to be true since we wish to see what the consequences of the properties expressed by these axioms are. Thus, any statement at all can be taken as one of many axioms. For instance, one may wish to study the natural numbers axiomatically by choosing no axioms at all. Well, without breaking eggs, you can't make an omelette, so all you'll get are tautologies. On the other extreme, you may take as axioms all the statements true about natural numbers. That will be a terrific achievement, but, of course, it is not realistic you can ever list your axioms in this case. The golden path lies in finding a clever, efficient, formulation of properties (of the natural numbers) that you deem to be characteristic, and state those as your axioms. So, in the modern context, the axioms of a system are simply an axiomatic description of some properties of whatever you wanna study (rather than an attempt to describe what things are, we describe what you can do with those things). For the natural numbers, the Peano axioms do the job.