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Eli5: The Transcendental Numbers?

What exactly The Transcendental Numbers are?
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Real numbers can be divided into two classes: Algebraic numbers, and transcendental numbers. All real numbers belong to one and only one of these classes.

Algebraic numbers are sort of the "nice" numbers. They are sort of the things that flow naturally from integers. All rational numbers are algebraic for example.

Transcendental numbers on the other hand are the murky, ugly majority of numbers that are difficult to deal with. Almost all numbers are transcendental, but they are exceedingly difficult to work with. We only relatively recently managed to prove pi belongs to this class of silent majority. 

The big thing about algebraic numbers is that they are solutions to equations like 2x-1 = 0. Or x^2 = 2. Solutions being x=0.5 and +/-sqrt(2). Basically, polynomial equations with integer coefficients. This covers big bunch of numbers we ever end up using. All integers, all rational numbers, square roots, cubic roots, etc. Transcendental numbers are "all the rest"

A thing to note is that all transcendental numbers are irrational, but not all irrational numbers are transcendental. For example, square root of 2 is irrational but algebraic.