Hi Trey - this is a very good topic and question
I'd like to share some basics about them and how I would approach composing/practicing those time signatures.
Let's take an example of 4/4. The first number here (one on top) indicates how many beats (or pulses) are within each bar of music. The second number, one bellow indicates which note duration defines one beat (pulse). So in the above time signature you have 4 x quarter notes in 1 bar of music. We usually count this time signature like this: 1-2-3-4
This means that you can fit notes in any durations in 1 bar of music as long as a sum of those notes doesn't exceed 4x quarter notes. Practical example would be playing 8x 8th notes in one bar of music instead of 4x quarter notes.
If we take the time signature of 3/4 it means there are 3 pulses of quarter note duration in one bar of music. You can't fit in this music measure (bar) 4x quarter notes as only 3 would fit this time. You can count this time signature like this : 1-2-3 ; 1-2-3....
5/4 would mean that you have 5x quarter notes in one bar of music. You can count it like this : 1-2-3-4-5 ; 1-2-3-4-5....
7/8 would mean there are 7x 8th notes in one bar of music. For better understanding you can see this as 3 quarter notes + 1 8th note as you are missing 1 eight note to get 4/4 time signature (or 8/8). You can count it in various ways, for example: 1-2-1-2-1-2-3 ; 1-2-1-2-1-2-3 where each number/count presents 8th note pulse. You can also simplify and just count to 7 like 1-2-3-4-5-6-7 but in 8th notes pulse (so faster than counting quarter notes before if you are playing the same tempo).
9/8 would mean there are 9x 8th notes in one bar of music. Same rules apply as above and count can be like : 1-2-3-1-2-3-1-2-3 ; 1-2-3-1-2-3-1-2-3 ...where each pulse is 8th note. You could also see it as 8th note triplets and count 1-2-3 ; 1-2-3 (3x 3 8th notes).
I have personally found myself always playing 4/4 time signature music with all my bands so I never got a chance to get used to other time signatures or dig deep but lots of progressive bands and especially metal ones tend to get more technical by using odd time signatures. Also it brings certain freshness to their music as 4/4 is so abused
Disclaimer : someone please correct me if I wrote something wrong up there about counting odd time signatures