Think of time signatures as counting systems. Have you heard a band leader count in a song like this, "1 and-a 2 and-a 3 and-a 4". He's counting off the four beats that the whole group will syncronize to. These groups of four are one of the many time-signatures.
Our example tune, Jingle Bells, is in the time signature of 4/4. This means that there are four beats in every section, (these sections are called measures, or bars), and that each quarter note gets one beat. See, the two numbers together mean 4 quarters.
If our tune was a waltz, then it would be in 3/4 time. This means that each measure has three beats, and that the quarter note gets the beat. Three quarters.
We can deduce from these examples that the top number of a time signature represents the beats in a measure, and the bottom number represents which type of note gets the beat, or the emphasis.
Other time signatures:
2/2, in which there are 2 half notes in each measure
5/4, in which there are 5 quarter notes in each measure
2/4, in which there are 2 quarter notes in each measure
3/8, in which there are 3 eighth notes in each measure
Get the idea?
Changing the number of notes in each measure also impacts the way a measure appears, and how you will count it. You would count 3/4 (waltz time) as 1 2 3, 1 2 3, 1 2 3 and so on.