The Pythagoreans believed all of nature was governed by ratios of whole numbers. It was a religious doctrine as much as a mathematical one. Then one of them — the legend names Hippasus of Metapontum — applied the Pythagorean theorem to the unit square and computed the length of the diagonal: √2. He attempted to express √2 as a ratio of integers. He failed. The proof that no such ratio exists is short and devastating, and the cult, according to legend, drowned Hippasus at sea to keep the discovery from spreading. Whatever the truth of the drowning, the mathematical fact is real and cannot be undone: not every length is rational. The diagonal of the unit square is the simplest irrational number, and there are uncountably more.
An irrational number is a real number that cannot be expressed as a ratio of integers. The classic proof that √2 is irrational: assume, for contradiction, that √2 = p/q in lowest terms (gcd(p, q) = 1). Squaring: 2 = p²/q², hence p² = 2q². Therefore p² is even, hence p is even, so p = 2k for some integer k. Substituting: (2k)² = 2q², so 4k² = 2q², so q² = 2k², so q² is even, so q is even — contradicting the assumption that p/q was in lowest terms. So no such representation exists, and √2 is irrational. The same technique shows that √n is irrational whenever n is not a perfect square. Other famous irrationals: π (proved irrational by Lambert in 1761; proved transcendental by Lindemann in 1882, settling the ancient question of whether the circle could be squared with compass and straightedge — it cannot); e (proved irrational by Euler in 1737); the Euler-Mascheroni constant γ ≈ 0.5772 (still not known whether rational, one of the better unsolved problems in number theory). Algebraic vs transcendental: an algebraic number is a root of an integer polynomial (so √2 is algebraic, satisfying x² − 2 = 0); a transcendental number is not (π and e). Algebraic numbers are countable (each polynomial has finitely many roots; there are countably many polynomials); the reals are uncountable; therefore almost every real number is transcendental, even though it is famously hard to prove transcendence for any specific number. The construction of the real numbers from the rationals — Cauchy sequences or Dedekind cuts — is the modern way to make irrationals rigorous: each irrational is identified with a specific equivalence class of rational sequences (or a specific cut of ℚ).
Irrationals are unavoidable in any geometry or analysis: the lengths of curves, the values of trigonometric functions, the solutions of differential equations — almost all of them are irrational. Computers approximate them with rationals (typically floating-point), and numerical analysis is the discipline of bounding the resulting error. Computable real numbers are those for which a finite program can output as many decimal digits as you want — they are countable, so almost every real is uncomputable (cardinality argument). Proofs of transcendence are a working frontier of number theory; Schanuel's conjecture would unify many open transcendence questions if proved.