4/4, 3/4, 2/4, 6/8, 12/8, these are just a few examples of time signatures. The top note refers to the number of beats per bar, while the bottom number refers to which note is given the beat. The 4 on the bottom, means a quarter note (crotchet) gets the beat, while 8 means eighth note (quaver) gets the beat.
Why Time Signatures?
4/4, or "common time", has 4 beats per measure. Why not count music pieces as, "1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,........." with no repeats?
Better yet, why count the beats of a piece of music at all?
The answer is because of accents.
Accents will open up the doors of understanding concerning rhythm. Each time signature has a certain location where Primary and secondary accents fall. In all time signatures, the heaviest accent falls on the first beat. Therefore, it is helpful when counting the rhythm of a song in 4/4 as "1, 2, 3, 4, "(the primary accents are indicated in bold.) So, in 4/4, the primary accent falls every 4 beats, while in 3/4 this accent falls every 3 beats and so on. You will notice that chords generally change around this accent.
Try Counting a song out for yourself and see if this isn't true.
There is also a secondary accent. This accent is not as heavy as the primary accent. It usually falls in the middle of the count. In 4/4, the 3rd beat is the secondary accent. (The secondary accent is shown in italics). Counting in 4/4, we have; 1, 2, 3, 4. In 3/4, the secondary accent falls on the 3 as well. In 6/8 the secondary accent falls on the 4.
Subdividing is simply counting more than once per beat. This is very helpful when a song has a slow tempo in order to prevent the tendency to rush. Instead of counting a slow song in 4/4 as, "1, 2, 3, 4", we may place 2 counts per beat, counting as, "1 & 2 & 3 & 4 &". We could also subdivide further, saying 4 counts per beat as, "1 e & a, 2 e & a, 3 e & a, 4 e & a". You could say anything you want between the beats, but this is the most common way people count out loud.