O--k-a-a-y. This could take a while...
Basically, it's something hard-wired into us. Our ears (and the part of the brain that interprets sound) work in such a way that certain sounds seem related. For instance, if you double the frequency of a sound, it seems to sound like the same sound on a different "level." This is what we call two "notes" being an "octave" apart. I'm sure you've heard that...
All right, then. Within an "octave," certain other sounds, um, well, sound "right," either together or in some relationship. I remember hearing a sound bite of Leonard Bernstein lecturing on this once--he pointed out the the "tune" you associate with (in English) the little-kid insult-chant "nyah, nyah" can be heard anywhere in the world. Those two "notes" seem to go together, and almost any human being will tell you so. And there are other "notes" that also seem to work well with those two.
You play with the math (and the sounds), and you end up with a "scale"--a set of notes that "go together" in different ways because they have a certain mathematical "family resemblance." Using those notes in the right sequence, you can trigger emotional responses in other people. It's a form of communication that is--partly--independent of words.
Now, if you're going to write all this down, or even agree with the guy next to you (so you can play your flute and he can play his harp and it will sound right together), you're going to have to arbitrarily pick some particular sound and say that's the "key" note. Once you have that note, the rest of them can be derived from it. We in the West have a particular set of sounds (and in particular, a single note) that everyone has agreed to treat as the standard, and all keys, chords, notes, etc., are derived from that standard.
So, sort of, a "key" is short for "the set of notes derived from THIS note to form THIS kind of scale."
Um, that's the other problem. Kinds of scales. Y'see, the emotional reaction you get from a set of notes in a song can be different depending on the exact notes in it. The two most obvious ones are the "major" keys (the classic "do-re-mi" stuff that most songs are written in if you don't think about it), which generally sound "happy" or "neutral," and the "minor" keys, which--again generally--sound "sad" or "thoughtful."
So you're talking about an arbitrary set of sounds, related in a way that isn't arbitrary at all to someone who hears them.
I hope this makes a little sense...