It seems strange to me that when we are asked to think about some numbers, we are most likely to think about numerals - especially Arabic Numerals $(0, 1, 2, 3, 4, ..... )$ which are merely some assigned symbols but what the 'Numbers' exactly are ?
In their most primitive form, the numbers are observed as something which we use to count so chronologically, one can say that, "Natural numbers are more fundamental than the numbers themselves", therefore we can shift our analysis to the Natural numbers rather than the wide spectrum of numbers which makes the goal somewhat attainable (in a single post). So before asking what natural numbers are, let's see what we know about them.
We know that Natural numbers are the numbers (not numerals) $0, 1, 2, 3, \dots $. Some people exclude $0$, some include $0$ and some do any one of both which is useful for their problem but we will assume it to include $0$. Similarly many things like we can use natural numbers as cardinals, ordinals and nominals but I'm more interested in defining Natural numbers from scratch rather than merely using them. An informal way to do so is to define a set
$$N= \{0, 1, 2, 3, 4, 5, \dots \}$$
and we can define Natural numbers as something (which is a set $N$) which starts from $0$ and then goes on indefinitely without the repetition . However it seems but it is not a good definition because of two reasons, first one being the method of circulation which we are using to define Natural numbers, we are using an advanced concept (of set) to define a fundamental concept (of $N$) but this advanced concept itself cannot be defined without using a fundamental concept, this is why it is creating a circulation and will be treated as an informal way to define the Natural numbers. Another problem is our assumption that the elements of the set $N$ are not repeating themselves which is absurd and the definition will said to be established on an absurdity if we define $N$ this way.
Let's now do something more concrete. Starting by some operations on Natural numbers like $5^3$ which is nothing but three times multiplication of $5$ by itself i.e. $5*5*5$ and again these multiplications are nothing but the $5$ times addition of $5$ times added quantity to itself:
$$5^3=5*5*5=(5+5+5+5+5)+(5+5+5+5+5)+(5+5+5+5+5)+ \\ (5+5+5+5+5)+(5+5+5+5+5)$$
It means the operation of $+$ is more fundamental which is defined as the incrementation in the Natural numbers.
$$ 5+3 \, \text{means the incrementation of 5 by three times}$$
But since we are trying to define the Natural numbers from scratch, let's forget about these numbers like $5, 3, 1$ and so on. Therefore, as we have a fundamental operation of increment, now we also need a fundamental natural number from which all other natural numbers will be pop out after applying the increment operation. Let's represent this fundamental Natural number by the symbol '$0$' and it's successor by '$0++$' and the next successor by '$(0++)++$' and the further successors in the same fashion. Now, we can use these two fundamental objects - '$0$' and '$+$' - to define the Natural numbers completely. And our Natural numbers are
$$0, 0++, (0++)++, ((0++)++)++, (((0++)++)++)++, \dots $$
To remove the burden of writing so much +'s, we can assign something shorter and simple to these Natural numbers with enormous +'s as follows :
$$ 1=0++ ; 2=(0++)++ ; 3=((0++)++)++ ; 4=(((0++)++)++)++ $$
and much more.
Finally our Natural numbers are the following 'objects'
$$0, 1, 2, 3, 4, \dots $$
We can conclude all this in terms of two Axioms which will define our Natural numbers $N$ (not as a set but a representation tool)as follows:
Axiom : 1 $0$ is a Natural number.
Axiom : 2 If $n$ is a Natural number then $n++$ is also a Natural number.
What now? Is the job done? Consider an example of a normal wall clock, it has Natural numbers from $1$ to $12$ ($0$ operated by $12$ $+$'s). The clock satisfies the Axiom 1 and 2 because it assigns the symbol '$1$' to the fundamental ($0=1$ here), there is no problem with the validity of Axioms in this clock, but if you ask her, "Hey! How would you describe the $N$?" Then it may reply, "Oh! I would like to define them as 12 objects i possess which are as follows:
$$1, 2, 3, 4, 5, 6, 7,8 ,9 ,10, 11, 12$$
(of course you can think about them as 1 with required increment operations). We can't object on this answer, it perfectly aligns with those two Axioms.
$$1\, \text{is a natural number and so are:}\, 2 = 1++ ; 3=2++ ; \dots ; 12=11++ ; 1=12++$$
so we learned from here that we need to introduce another Axiom to eliminate the circulation
Axiom : 3 For any Natural number $n$, $0 \neq n++$
But we haven't solved the problem yet. Now consider an abnormal wall clock which stops each time after one of its hand get vertically up (to point the '$12$'). If you ask the same question to this clock, it will give you the same answer with the following logic:
$$1\, \text{is a natural number and so are:}\, 2 = 1++ ; 3=2++ ; \dots ; 12=11++ ; 12=12++$$
(Instead of assigning the fundamental to '$12++$', it will be frozen at the last Natural number and all three Axioms are still valid)
And you may notice that Axiom 3 solves the problem of a particular circulation only where the final Natural number makes a loop when connected to the fundamental 0 but it doesn't say anything if we define any natural number let's say $4$ as $4++=1$ or $6++=2$, they are also creating circulation so we need an Axiom which deals with more general circulations. This causes us to have the following Axiom:
Axiom : 4 If we have two natural numbers $m$ and $n$ such as $m++ = n++$ then $m=n$.
Now it seems the job is done. These four Axioms are sufficient to define $N$. If we include one more Axiom not to define $N$ but as something which defines the properties satisfied by the $N$, then we would have a special collection of Axioms.
Axiom : 5 If $P$ represents a property and this $P$ is true for $0$ i.e. $P(0)$ is true and if we assume $P(n)$ to be true for some arbitrary $n$ which causes $P(n++)$ to be true for each such $n$, then we say $P$ is true for all $N$.
This Axiom is so important that it got a special name 'Principle of Mathematical Induction'.
Now these 5 Axioms collectively are known as 'Peano Axioms'
