A number is transcendental if it is not the root of any polynomial equation with integer coefficients. pi satisfies no equation like x^2 - 3x + 1 = 0. e satisfies no such equation. They exist beyond the reach of algebra. Despite being rare to name, transcendental numbers are the rule rather than the exception: almost every real number is transcendental.
Every rational number is algebraic. Every algebraic number is real. But the transcendentals, the numbers outside the algebraic ring, are vastly more numerous than all algebraic numbers combined.
The 1882 proof that pi is transcendental finally settled squaring the circle: a 2000-year-old geometric challenge proved impossible because you cannot construct transcendental lengths with compass and straightedge.
A number is transcendental if it satisfies no polynomial equation with integer coefficients. Liouville gave the first explicit example in 1844. Hermite proved e is transcendental in 1873. Lindemann proved pi is transcendental in 1882, finally settling the ancient squaring-the-circle problem as impossible. The Gelfond-Schneider theorem (1934) shows that a^b is transcendental whenever a is algebraic and not 0 or 1, and b is algebraic and irrational. Despite being the rule rather than the exception, proving any specific number transcendental remains extremely difficult.