A real number is an approximation of a rational number. (This is if you use the usual ordering of rational numbers; others will give the idea of -adic numbers?.)
The original idea of a real number came from geometry; one thinks of a real number as specifying a point on a line. (More precisely, given two distinct points on the line, called and , you get a bijection between the points and the real numbers.) Euclid? (citing Eudoxus?) dealt with ratios of geometric magnitudes, which give positive real numbers; an arbitrary real number is then a difference of ratios of magnitudes. (But the Greeks did not think of such ratios as numbers.)
A big project of the 19th century (at least in hindsight) was the ‘arithmetisation of analysis’: showing how real numbers could be defined completely in terms of rational numbers (and the desired classes of functions on them could be defined in terms of the general point-set notion of function). Two successful approaches were developed in 1972, Richard Dedekind?'s definition of real numbers as certain sets of rational numbers (called Dedekind cuts) and Georg Cantor?'s definition as certain sequences of rational numbers (called Cauchy sequences).
A more modern approach is instead to characterise the properties that the set of real numbers must have and to prove that this is categorical (unique up to a unique bijection preserving those properties). Then the important result of the 19th century programme is simply that this is consistent (that there exists at least one such set). One can even use Hilbert's or Tarski's axioms for geometry to do this characterisation, coming full circle back to geometry.
Exactly how to define or characterise real numbers is still important in constructive mathematics and topos theory. We should consider possible definitions (including those that don't make the numbers primary) and their consequences below, but I'm done writing for now.