Pi is defined as the circumference of a circle divided to its diameter, and it's the best known irrational (and transcendental) constant that's known since antiquity. It shows up when working with circles and trigonometric ratios and other geometric functions, as well as some other places you wouldn't expect to. It is considered a cult number for this reason, and many people calculate or memorize the digits just for the sake of it.
Pi is an example of a transcendental number, a number that cannot be the solution x for P(x) = 0 where P(x) is a non-zero integer polynomial of x. The numbers that can be the solution of such equations are called algebraic.
Pi is sometimes mistakenly used in googology as an attempt to trounce others, since pi's decimal digits go on forever; but pi is smaller than 4, so it can be easily beaten. A better attempt would be "pi with the decimal point removed" but that leads to an endless decimal expansion which does not equal to any finite number. A rare legitimate appearance of pi in googology is on Stirling's approximation on large factorials.