What exactly is real number?

Basically, mathematicians don't care at all what a mathematical object is, we only care about what we can do with it (what operations are defined and what are their properties). So one mathematician might construct real numbers as equivalence classes of Cauchy sequences of rationals, another might prefer Dedekind cuts. Since there is a one-to-one correspondence between those sets of "real numbers", preserving all the structures that we want to define on the real numbers, the disagreement between the two is inconsequential.

I would like to add two things to the discussion:
1. First of all, if you consider standard sets of numebrs, such as natural, integer, rational, real and finally complex number, being very formal, you don't have ANY of inclusions $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$. Each such inclusion should be understood as a natural embedding: integers are equivalence classes of pairs of natural numbers, subject to relation: $(m,n) \sim (p,q)$ iff $n+p=m+q$ (then the embedding is $n \mapsto [(n,0)]$), a rational number of the form $\frac{p}{q}, q \neq 0$ is an equivalence class of pair of integers (the second integer is assumed to be nonzero), subject to the relation $(m,n) \sim (p,q)$ iff $np=mq$ (then the embedding is $k \mapsto [(k,1)]$), any real number is the equivalence class of some Cauchy sequence of rationals (as explained in the answer above: the embedding is $q \mapsto [(q,q,q,...)]$) and any complex number is a pair of real numbers (with the natural embedding $x \mapsto (x,0)$).
2. As you may have already noted, the most dramatic is the passage from rational numbers to the reals: to define integer you need a pair of natural numbers, to define a rational number you need a pair of integers, to define a complex number you need a pair of reals but to define a real number you need infinitely many rationals. As a result of this, the set of all real numbers has no longer the same cardinality as the set of rational numbers. But the main property which is new is the completeness of the reals: roughly speaking you cannot do any analysis without this property so the countability is the price that you pay in order to do any serious analysis.

A real number can either be accepted as a primitive notion, or defined in terms of "simpler" primitive notions. A canonical method constructs the real numbers from the axioms of ZFC set theory. You can read more about this in Enderton's Elements of Set Theory.

Ultimately, this is a question about the philosophy of mathematics, and there are several common doctrines, including the one you mentioned about the "very nature" of mathematical objects.