Short answers, pulled from the story.
A real number is a number that can measure a continuous one-dimensional quantity such as a length, duration, or temperature. Every real number can be almost uniquely represented by an infinite decimal expansion, and real numbers include both the rational numbers and the irrational numbers.
René Descartes introduced the adjective real in the 17th century to describe the roots of a polynomial and to distinguish such numbers from imaginary numbers, like the square roots of negative numbers.
Rational real numbers, such as integers and fractions, can be written as a ratio of two integers, while irrational real numbers cannot. The square root of 2 is irrational, a fact the Greeks under Pythagoras realized around 500 BC.
Dedekind completeness states that every non-empty set of real numbers with an upper bound has a least upper bound. This property distinguishes the real numbers from the rational numbers, since the rationals below the square root of 2 have no least rational upper bound.
The set of all real numbers is uncountable, meaning no one-to-one function maps the reals to the natural numbers. Cantor showed in 1874 that the reals are uncountably infinite while the algebraic numbers are only countably infinite.
Two independent definitions were published in 1872, one by Richard Dedekind as Dedekind cuts and one by Georg Cantor as equivalence classes of Cauchy sequences. Dedekind had begun this work in 1858, and Cauchy had already made calculus rigorous in his Cours d'Analyse of 1821 without defining the reals.
Finite computers cannot store infinitely many digits, so they use finite-precision approximations called floating-point numbers, often a 64-bit representation with around 16 decimal digits of precision. Because there are only countably many algorithms but uncountably many reals, almost all real numbers are not even computable.