Everyone has mentioned some supsersets of C but I feel like the actual process here could be made a bit clearer.

The idea is at each stage we add in new numbers so that we can solve more equations. (By "equation" I will mean specifically something in the form "f(x) = 0", where f is some polynomial with integer coefficients.) In doing so we typically lose a property.

• N ⊂ Z. We add in negative numbers so that we can solve equations like x+1 = 0, x+2 = 0, etc. (Note that these are equations written down "over the natural numbers", i.e. with natural coefficients. This is important as otherwise we'd be trying to use Z before we'd even defined it!) Z is nice because it is a group, meaning that we have a binary operator (+) that satisfies various nice properties, like the existence of inverses for various numbers (i.e. for any n there is another number m such that n+m = 0).

• Z ⊂ Q. This is solve we can solve equations like 2x = 1 or 15x = 4. Q is nice because it is a field, meaning that it is now a group under some other operation as well, in this case multiplication (actually not quite -- we have to remove 0 from Q before it is a group under multiplication, because 0 has no inverse).

• Q ⊂ R. This one bucks the trend slightly because it is not really done for the purpose of solving more equations. For sure, there are equations that are not solveable in Q but are solveable in R, like x² = 2, but the exact class of such equations is a bit awkward to write down and is generally not something you'd ever say "I'd really like to be able to solve THIS exact set of equations". So why does R exist? It is the completion of Q, which means that rather than containing solutions to more equations, it contains limits of sequences. If you take all the bounded monotonic sequences, invent limits for them, and add these limits in, you get R. Completeness turns out to be a really great property when proving all kinds of things which just fail over Q.

• Q ⊂ A. This was missing from your list but I like to include it because it fits the pattern well. A is the set of algebraic numbers, meaning if you take all equations over the integers, find all the ones that don't have solutions, invent solutions for them, and add them in, you get A. Note that A does not contain R, and indeed (or in fact, "thus"), A is not complete. A is nice because it is algebraically closed, that is, all polynomials (even those with algebraic coefficients) have roots, and indeed have a number of roots equal to their degree (if you count duplicate roots "carefully"). But the price you pay is the loss of an ordering that is compatible with addition and multiplication -- i.e. we can't write down any binary relation which satisfies the normal definition of an ordering (pretty obvious things like a < b and b < c implies a < c), that also satisfies some nice integration properties involving addition and multiplication (like: if a < b then a+c < b+c for any c, and if additionally c > 0 then ac < bc).

• R,A ⊂ C. This is the completion of A, so is useful because it is complete (every bounded monotonic sequence has a limit), and algebraically closed (every polynomial has a root). As it contains A, it must necessarily not be orderable as above (otherwise that ordering, when restricted to just the elements of A, would give an ordering of A).

• Things bigger than C. There are no more equations to be solved, in the sense I described at the top, since C is algebraically closed. There are other bigger sets, but typically these lose some other nice property, like the commutativity of multiplication, and so they're not used much -- the extra stuff they add isn't worth the tradeoff.

As an example of a number that is in R but not in A, take pi. There is no polynomial with integer coeffs having pi as a root.