This is a music theory question: in theory, what is the difference between time signatures? Some signatures are only pedantically different: essentially they are the same. But rhythm is based in emphasis, and that is what decides the signature.
Think this way - Each measure, we will typically have 2, 3, or 4 beats. If we had 1 beat per measure, it wouldn't define a measure, as the measures would sound like one continuous pulse with no emphasis. That isn't generally considered musical.
In addition to the number of beats, we can also have the divisions of the beats. If we divide into two or four parts (four parts being two parts divided twice), we get a simple BA-dum BA-dum, but if we divide into three beats we get a more complex sound, BA-dum-dum BA-dum-dum.
Under these regular beat conditions, there are only 6 time signatures.
The first three are the simple meters:
Common time, simple quadruple (4/4 or c) is standard. The hardest emphasized beat is 1, and the following beats are a soft 2, a medium hard 3, and a soft 4. Then it repeats. This sounds usual.
Simple triple is 3/4, or waltz time. This is the rhythm to "Happy Birthday" and many famous pieces, and although distinct it is also pretty usual. The only hard beat is 1, 2 and 3 are softer and often arpeggiated.
Simple duple is 2/4. It isn't exotic but it isn't used as much because it often sounds like 4/4. 2/4 is easiest to think of like 2/2, which is a marching cadence beat. But 2/2 is technically like 4/4 ... the difference is often just what style of music is being played.
Note that in the simple meters, we only have the BA-dum as sub divisions, 8th notes and 16ths. Any other subdivisions, like triplets, are considered irregular.
In complex meters, we have the three beat subdivisions, almost like a mini-triple measure exists in every beat (actually, if you play 3/4 fast enough, say 160 bpm or higher, it sounds like a complex meter). This means that other divisions, such as duples, are irregular in complex meters. We can mathematically make them from the simple meters by multiplying the signatures by 3 subdivisons/2 subdivisions, 3/2.
6/8 is the complex duple.
9/8 is the complex triple.
12/8 is the complex quadruple. This meter is almost always chosen by someone when they want to do both duple and triple subdivisions - this is because 12 is divisible by both 2 and 3, and in such a way that divisions land on the beats (6/8 doesn't have this property, because the duple divisions are on upbeats). This effect of 2 against 3 is known as hemiola.
There are also irregular meters, like 5/4 and 7/8. These meters are defined additively, i.e. 5/4 would be either a 3 / 4 + 2 / 4 or 2 / 4 + 3 / 4, depending on where the hardest emphasized beat lands.
Don't listen to 7/8 songs unless you are really weird, and don't play them unless you are really good. It is is easy to add an 8th to 7/8 and slip into 4/4, or subtract one and slip into 3/4. That awkward place in between the two very distinct signatures is 7/8.
A 5/4 song example is the Mission: Impossible Theme, although sometimes it is 4/4 too, so that would give you a comparison.
There are also mixed meters, where the measures regularly change meters. So if I decided to do 3 measures of 4/4 and one of 3/4, that would be mixed. The pop song Hey ya is a brilliant example of this.
Maybe a bit more mathematical then you hoped for, but that's where they come from. All about emphasis, not about reading generally.