Transcendental numbers are a strict subset of the irrational numbers.
Although "almost all" real numbers are transcendental it is difficult to prove that a given number that occurs naturally is transcendental.
The size of the set of transcendental numbers is the size of the continuum c. (See Uncountable sets).
Some numbers shown to be transcendental are $e$ , $\pi$ , sin(1) , log(2) and $e^\pi$
Whether $2^e$ , $2^\pi$ or $\pi^e$ are transcendental or not has still to be decided.