As a piano teacher, I have been asked this question many times and sometimes the information available is not clear cut.
First of all, every time signature that exists in western traditions of music falls into the category of either simple time or compound time.
Simple time includes: 2/4, 3/4, 4/4
Compound time examples are: 6/8, 9/8, 12/8
These are the most basic time signatures and the probably account for ninety five percent or more of all piano music.
These following time signatures range from unusual to rare and they include:
Simple time: 2/2, 3/2, 4/2, 2/8, 3/8, 4/8,
Compound time: 6/4, 9/4, 12/4, 6/16, 9/16, 12/16,
In order to understand what the difference is between simple time and compound time, it is important to understand what the bottom number means, as well as the top.
The lower number refers to what is used as the beat so;
2 refers to a minim or half note beat
4 refers to a crochet or quarter note beat
8 refers to a quaver or eighth note beat and
16 refers to a semiquaver or sixteenth note beat
The top number in a time signature refers to how many beats there are. In terms of working out whether it is simple time or common time, depends on how the notes are grouped. It is important to understand how the notes are grouped in order to understand the fundamental rhythm or pulse of the time signature.
One way to remember is that 2/4, 3/4 and 4/4 are simple time. 2/2, 3/2 and 4/2 are the same except their beat is equal to a minim or half note, but it would have the same sort of feel to it. Likewise 2/8, 3/8 and 4/8 is also the same feel except the quaver or eighth note is the beat. Anything with a time signature that the top number is a 6, 9 or 12 is going to be compound time because the beats or pulses are grouped in threes, which gives a very different beat or rhythm to the simple time signatures.
Sometimes people get confused with 3/4 and 6/8 because essentially they both have the equivalent of 6 quavers or eighth notes in each bar or measure. With 3/4, the quavers are grouped in three groups of two quavers or eighth notes and 6/8, the quavers are grouped into two groups of three quavers or eighth notes.
Another way of looking at it is:
3/4 1 1½ 2 2½ 3 3½
Pulse Strong very weak weak very weak weak very weak
6/8 1 2 3 4 5 6
Pulse Strong weak weak medium week week
As you can see, 3/4 and 6/8 are significantly different, not just with these two time signatures but with understanding the difference between simple and compound time.
It may seem a bit complicated at first, but getting straight what simple time and what compound time it important. Hopefully this will have helped.