For variety, I will also give another construction.
From the rational numbers, you apply the ideas of non-standard analysis to construct the hyperrational numbers. You can then construct the real numbers as the standard parts of limited hyperrational numbers.
- Any limited (i.e. not infinite) hyperrational number is the name of a real number
- Every real number can be named in this way
- Two limited hyperrational numbers are both names for the same real number if and only if they differ by an infinitesimal.
EDIT: to answer the question of the post, from one point of view, central idea behind the constructions is that the real numbers fill in the "holes" between rationals -- i.e. the real numbers are a continuum whereas the rationals are not. The method I describe here and the two classic ones Henning mention are all schemes for locate the holes. If you know where the holes are, then you know what the irrational numbers need to be, and so you can define the reals in that fashion.
The difference as compared to constructing the integers and rationals from the natural numbers is, in some sense, how much data is needed to identify a real number.
Identifying integers as things that ought to be the difference between natural numbers only requires two natural numbers to identify the integer. Similarly for the rational numbers as quotients of integers.
The classic constructions using Dedekind cuts or Cauchy sequences use an infinite amount of data to pick out a real number. Since the data can be chosen in a sufficiently arbitrary way, the number of ways to select data is a greater cardinal number than the number of rationals.
As for the version I describe, only two hyperrationals are needed to locate a real, but the hyperrationals are already (externally) uncountably infinite.