The important sets of numbers are characterized by the fundamental operations we expect to be able to do with them. This is also called "closure" of an operation; for example way say that $\mathbb{Z}$ is closed under subtraction. In most cases, constructing these sets from set theory axioms is just aimed at making those operations possible.
The natural numbers $\mathbb{N}$ is a set where we can:
- Have a unique smallest number (either $0$ or $1$ depending on context and discipline).
- Take the unique next number after another number (successor operation).
- Take the smallest number in any non-empty set of numbers.
- As a consequence of the above properties, we can also add and multiply numbers in $\mathbb{N}$.
($\mathbb{N}$ is actually a fairly nuanced and interesting case! But it's not the focus of this question or answer.)
The integers $\mathbb{Z}$ is a set including $\mathbb{N}$ where we can:
Here "including" means for every day use we can think of the relation as $\subset$, but in the set theory definitions this is not true, and we instead have an injective function which preserves the operations like addition which make sense in both sets.
The rational numbers $\mathbb{Q}$ is a set including $\mathbb{Z}$ where we can:
- Add.
- Subtract.
- Multiply.
- Divide, when the divisor is not zero.
The algebraic numbers $\overline{\mathbb{Q}}$ is a set including $\mathbb{Q}$ where we can:
- Add.
- Subtract.
- Multiply.
- Divide, when the divisor is not zero.
- Take the roots of any polynomial with coefficients in $\overline{\mathbb{Q}}$.
Although it's quite common to discuss polynomial roots in $\mathbb{C}$ or $\mathbb{R}$, those sets actually have properties not required for that purpose and "extra" numbers which are not roots of polynomials over $\mathbb{Q}$ (i.e., transcendental numbers).
The real numbers $\mathbb{R}$ is a set including $\mathbb{Q}$ where we can:
- Add.
- Subtract.
- Multiply.
- Divide, when the divisor is not zero.
- Take the limit of any converging sequence.
The complex numbers $\mathbb{C}$ is a set including $\mathbb{R}$ (and, it turns out, $\overline{\mathbb{Q}}$), where we can:
- Add.
- Subtract.
- Multiply.
- Divide, when the divisor is not zero.
- Take the limit of any converging sequence.
- Have a number $i$ where $i^2 = -1$.
- Using the above properties, we can prove the Fundamental Theorem of Algebra, meaning we can take the roots of any polynomial with coefficients in $\mathbb{C}$.
Note each time we add a new operation type, the set theory development will first define the new number set in terms of the operation's action just on an existing number set. Then we can also prove a similar operation makes sense on the entire new sense in a way consistent with the first limited operation, and without needing any additional numbers in the set (this is the "closure" property again). For example, $\mathbb{Z}$ is usually defined based on pairs in $\mathbb{N} \times \mathbb{N}$ with equivalence classes inspired by the idea of subtracting any two natural numbers; then we can prove we can also subtract two of these new integers. Likewise, the set theory definitions of $\mathbb{R}$ use rational numbers as the elements of Cauchy-convergent sequences or Dedekind cuts, and then we can prove limit properties in $\mathbb{R}$ more generally.
So I'd say the "everyday" definitions of these sets are just the fundamental properties, and a few of the most important theorems, listed above. Related to that are the important rules about the operations themselves, like $x+y=y+x$ and $x \cdot 1 = x$ and $(x+y)-y=x$, but I think of that as more "what addition, subtraction, and multiplication are", beyond the fact that "integers are numbers we can subtract". For everyday use we're not really concerned with how set theory can define the sets of numbers and their operations and show that the operations satisfy our expected rules, just that the numbers and operations are there.
From this point of view, the essential thing the real numbers have when compared to the rational numbers is closure of converging limits. (We do need to specify "converging", just to rule out unbounded sequences which don't converge at all.) A decimal representation (or other base $b$ representation), Cauchy-convergent sequences of rationals, and Dedekind cuts are all ways of adding this concept and operation to $\mathbb{Q}$. When study of geometry suggests we need numbers with properties like $x^2 = 2$ (which we can give names like $\sqrt{2}$), the intuition such numbers "ought to exist" probably reasons from the fact that the square of a rational number can get arbitrarily close, but never exactly equal, to $2$, and again the limit concept comes into play. So introducing the set $\mathbb{R}$ makes sense, although the other set $\overline{\mathbb{Q}}$ would also suffice for finding square roots. Again, every day use doesn't need to care about how real numbers are defined in set theory, just that the numbers and their properties including sequence limits are there.
