First, let me go over what a time signature is for the non-musicians reading this. At the beginning of a piece of music, there are several different symbols including a couple of numbers stacked on top of each other like a fraction. Sometimes there is a large letter C or a C with a line through it instead. This is the time signature.
|C with a line means the same as 2/2.|
|C and 4/4 are also the same. Don't worry about why, they just are.|
What the time signature really tells us is what the musical note values in one measure will add up to. That's it, nothing else.
Simply put, the numbers are the fraction of a whole note within each measure. So 3/4 means there is three fourths of a whole note in a measure which is the same as 3 quarter notes. However, a 3/4 time signature is more likely to use the dotted half note for the beat than the quarter note. Another example is 6/8 which means there are six eighths of a whole note in one measure which is the same as 6 eighth notes. And the 6/8 time signature rarely uses the eighth note for the beat; the dotted quarter note is a much more common choice with this time signature.
A more complex way to say this is: The bottom number represents a note value. This means 2 is a half note, 4 is a quarter note, 8 is an eighth note, 16 is a sixteenth note and so on. The top number tells you how many of the note values represented by the bottom number are in one measure. So 4/4 means there are 4 quarter notes in one measure and 3/16 means there are 3 sixteenth notes in one measure. Now 4/4 sometimes uses quarter notes for the beat but just as often uses the half note or the sixteenth note depending on how fast or slow the piece is overall. In 3/16 the sixteenth note, the dotted eighth note or even the 32nd note can be counted as the beat. There is simply no way to tell from the time signature. It also doesn't tell you what rhythm patterns or meter will be used. These things are often implied by the time signature but there is no absolute relation between time signature and which note you count or the meter used. The music itself is a much better guide for figuring out the meter. And the speed (with the meter in mind) most often determines what note will be counted. It is a good idea to match a time signature with the meter in some way that people can understand but it is quite possible to impose unexpected time signatures on any meter if you are stubborn enough and don't mind making your music very difficult to play. Stravinsky did this at times, apparently in an attempt to drive his orchestra to distraction. But in general, we like our meter and time signature to work together in some way.
In a sense, the only thing the time signature reliably tells us is where to place the bar lines between each measure. This is very important for keeping your place in the music, especially when there is more than one musical part in a piece. Keeping a musical group together without a time signature or bar lines is a much more complicated process!