The most consequential fact about the square is that its side and its diagonal cannot both be measured by any common unit, however small that unit is made. The diagonal is sqrt 2 times the side, and sqrt 2 is irrational.
The classic proof is short. Suppose sqrt 2 equals a fraction in lowest terms, p over q. Then p squared is twice q squared, so p squared is even, so p is even, so p squared is divisible by four, so q squared is even, so q is even. Both being even contradicts the fraction being in lowest terms. A version of this argument appears in Aristotle, and it circulated as an appended proposition in the Elements.
The discovery is traditionally credited to the Pythagoreans in the fifth century BC and is usually described as a crisis for them, since their programme took the world to be intelligible through ratios of whole numbers. The story that Hippasus was drowned at sea for revealing it is repeated constantly and is not reliable: the sources are centuries late, they disagree over whether the offence concerned irrationality or the dodecahedron, and some present the drowning as divine punishment rather than as murder. Historians of mathematics treat it as legend. That incommensurability was known and that it mattered is not in doubt; the melodrama around it is.
The real effect was on method. Greek mathematics after this point handles magnitude geometrically rather than numerically, and Book V of the Elements sets out the theory of proportion credited to Eudoxus, which treats ratios of incommensurable magnitudes rigorously without ever needing a number for them. It is a workaround of extraordinary quality, and it held for close to two thousand years, until the nineteenth-century construction of the real numbers.
Plato's Meno turns the same fact into a scene. Socrates asks an untutored slave boy to double a square, watches him guess that doubling the side will do it, and leads him to see that the answer is the square raised on the diagonal.