I've read the Wikipedia articles for both procedural programming and functional programming, but I'm still slightly confused. Could someone boil it down to the core?
A functional language (ideally) allows you to write a mathematical function, i.e. a function that takes n arguments and returns a value. If the program is executed, this function is logically evaluated as needed.1
A procedural language, on the other hand, performs a series of sequential steps. (There's a way of transforming sequential logic into functional logic called continuation passing style.)
As a consequence, a purely functional program always yields the same value for an input, and the order of evaluation is not well-defined; which means that uncertain values like user input or random values are hard to model in purely functional languages.
1 As everything else in this answer, that’s a generalisation. This property, evaluating a computation when its result is needed rather than sequentially where it’s called, is known as “laziness”. Not all functional languages are actually universally lazy, nor is laziness restricted to functional programming. Rather, the description given here provides a “mental framework” to think about different programming styles that are not distinct and opposite categories but rather fluid ideas.