In simple English, what does it mean for a number to be “transcendental”?

A real number that's not algebraic is called transcendental. While fifty upvotes on some stack exchanges is not a large number, on Math SE fifty is a lot; less than 0.2% of answers score so high. Okay, transcendental numbers have something to do with polynomial equations, but why do we care about polynomial equations? It's just explaining one obscure mathematical abstraction in terms of second one. Suppose you have some number !!x!! You start with !!x!! and then you can add, subtract, multiply, or divide by any integer, except zero. Numbers like !!pi!! with which you can't win are called transcendental. Transcendental numbers are fundamentally different: they are not arithmetically related to the integers via simple steps. One often sees explanations of "Irrational number" that refer to the fact such a number has a nonrepeating decimal expansion.