One of the problems with getting older is that you have to unlearn virtually everything you were taught when you were young. This is true of every subject, and when it comes to music, a good deal of the bad intel centers about rhythm.
Nearly everyone that I know who has had piano lessons learned very early on that 4/4 time means that there are four beats in a measure and a quarter note gets one beat. This last bit of information was really pretty useless as you probably spent the first year and a half playing music in 4/4 time or 3/4 or 2/4, so that the bottom number never changed at all--it was always and forever a 4, and thus knowing why it was there was unnecessary and more than likely you completely forgot about it.
It is also, unfortunately, wrong. If and when you got around to playing a piece of music in 6/8 time, you might have assumed that there were six beats in a measure and the 8th note got the beat. That is simply to apply the same logic to the explanation of 4/4 above. But you would be wrong, because most of the time, when someone is performing a piece of music in 6/8 time, there are really two beats in a measure, so that the eighth notes by themselves are only parts of a beat.
The rhythmic universe is actually pretty simple, but only if you view it from the right angle. Pretty much everything in music can be split into either groups of two or groups of three. A time signature with an 8 on the bottom is likely to have a multiple of 3 on the top, and the eight notes are going to be grouped into sets of three, added together to make one beat. A time signature with a 4 on the bottom is going to be counted in groups of two eighth notes, which together form a beat. Thus time signatures are either simple (with a 2 or a 4 on the bottom) or compound (with an 8). On the page you can see the difference by whether eighth notes are grouped into set of two or three, and whether there are dots after the long notes, making them all divisible by three.
We call these two kinds of meter simple and compound. 4/4 is simple meter, and would be counted
1 and 2 and 3 and 4 and
whereas its compound meter cousin, 12/8, would go like this:
1 and uh 2 and uh 3 and uh 4 and uh
(if you counted each eighth note that would be 1 2 3 4 5 6 7 8 9 10 11 12, but as you can hear, that's a bit cumbersome; hence the approach above, where the second and third eighth notes or a group lose their numbers and become simply ands and uhs)
That's pretty much all there is to it, except for mixed meter, which is a combination of the other two. A bar of 5/8, for instance may feature a group of two eighth notes followed by a group of three. Before mixed meters came into prominence in the last century there was very little fraternizing between the worlds of simple and compound meter; it was either groups of two or groups of three, never both in the same measure, or for that matter, even the same piece. If that rule was ever violated, it was announced with a special number above the unusual grouping (like the 3 above a triplet). Also, once the time signature was announced at the beginning of the piece, it didn't change. That's not always true anymore, and there is where things can get difficult, especially if you don't have a clear idea of what time signatures really mean.
What time signatures look like on the page and how they are performed may be two very different things, as we've already noticed. 12/8 doesn't mean there are twelve beats. 5/8 doesn't mean there are five. What it really means is an indication of rhythmic real estate. How much space exists in each measure? In the case of 12/8, you can get 12 eight notes, or the equivalent (like 6 quarter notes) into one measure of music. That's all. The pulse is an entirely different matter, and it may not be represented in writing at all! The time signature doesn't tell you how to feel the piece.
So what's up with that, anyhow?
A lot of folks would like to think that one sunny day in the storied past, a nice old chap with nothing to do that afternoon sat down and single-handedly drafted the entire system of rhythm that we have today, and, since he got to define every bit of it himself and since he was obviously a bright fellow, everything should be consistent and easy. But that's not what happens, ever.
Instead, various people try various things, and what eventually happens is a collision of various systems governed by various rules, which may in themselves be consistent, or have made sense at one time, but now we're stuck with them.
Exhibit a: the English language. How is it that the letter f is pronounced like an f, and the letters gh are also (sometimes) pronounced like an f, and so is ph, although sometimes gh is actually silent and of course, c and k should get downsized because at least one of them is redundant, I mean, S can do the job in the first part of the word circle, and k can do the rest and don't get me started on homonyms.
Basically, what's happened is that different groups of people, speaking various languages, have had useful words exported to English, and some of those different rules for pronunciation, including some of their letters, went with them. Usually they got corrupted along the way. It has led to a confusing set of exceptions to the rules that we were taught as children--first the rules, then the exceptions, which nearly outnumber the rules. And nobody really has the authority to clean it all up and start over.
By contrast, musicians have a much easier road. For the real definition of a time signature, simply as a measure of space, and not as an indication of number of beats or who gets the beat or whatever, has no exceptions. It does have the disadvantage of not telling us very much, but at least it is never wrong. 16/8, you want to know? Why, you can get 16 8ths notes in there. Don't see that one very often. how about 5/2? Five half notes, or the equivalent. 19/3? Doesn't exist. There is no 1/3 note in music.
Remember how that bottom number is supposed to represent a note value? Some 20th century composers began writing time signatures with a note on the bottom instead of a number, so that 4/4 would have a 4 on top and a quarter note on the bottom. 12/8 would have a 12 on the top and an eighth note on the bottom, although that was often replaced by a 4 on the top and a dotted quarter on the bottom, which is a truer indication of the way you would actually play the piece, and finally makes your piano teacher sound like she's telling the truth--the number on the bottom at last really does function as an indication of what note gets the beat. No word yet on whether this is going to catch on universally. My money is on forget about it. Numbers are too entrenched, as are many customs, long after we have the slightest clue what they are about.