The textbook answer I always received was, "The top number tells you how many beats per measure, and the bottom number tells you what value note receives one beat." So 3/4 time means "three beats per measure, and a quarter-note receives one beat." That tells you that each measure will contain three quarter-notes or the equivalent thereof.
From my point of view as a hobby musician (I don't read music very well), I look at it this way: The top number gives you the rhythmic feel of the song, which is generally the most important thing to know. The top number can (theoretically) be any integer, but the most common ones are 2, 3, 4, 6, 8, and 12. Occasionally you'll see other numbers, such as 5, 7, and 9. There's a Wikipedia article that lists lots of songs with unusual time signatures, with "unusual" most often defined in terms of the top number.
The bottom number is more about how the composer chose to write the actual sheet music, in conjunction with the feel given by the top number. If the composer found it easier (or more explanatory) to write most passages using quarter-notes, the bottom number will more likely be a 4. If eighth-notes better suit the composition, the bottom note will be an 8. It's largely a judgement call. Sometimes composers will write in unusual time signatures as an exercise, and sometimes an unusual time signature will be used to add an extra beat (or two or more) to a phrase. Both of these cases can be seen in the above-linked Wikipedia article.
Often I will try to reverse-engineer a song without referring to the original sheet music. The most frequent variable I've encountered when doing so is whether the bottom number is a 2 or a 4, but I've also found some variability between 4 and 8. There can also be ambiguity about the top number: I've sometimes thought a song was in 2/4 when it turns out to have been written in 4/4. I've also sometimes thought a song was in 3/4, 6/8, or 12/8, only to find that it's written in X/4 time, with lots of triplets. (A triplet is when a note's time is divided into three notes, usually written as three barred eighth-notes with a small number 3 above or below the bar. Think of the "merrily, merrily, merrily, merrily" part of "Row, Row, Row Your Boat.")
But another very important factor is something the OP didn't ask about: the tempo notation. On sheet music you'll often see, at the top of the piece, a quarter-note followed by an equal sign followed by a number, as shown here:
This gives us very firm guidance on the tempo of the song: 90 quarter-notes (or 90 beats) per minute. You can set that number on a metronome and you'll know that you're at the tempo the composer wanted. But nearly as often, you'll just see a verbal description of the tempo, such as "Moderately," "Brightly," "Quickly," or some such. Most classical compositions use Italin words to describe the tempo, such as "Andante," "Presto," "Lento," and so forth. Classically trained musicians develop a feel for what each of these means, but there are also guidelines available through a simple online search.