Time signatures in music:
The top number of the fraction are how many beats make up a measure. The bottom number is what kind of note gets a beat. It’s all based of the European music standard of 4/4 - four beats per measure, and each beat is a quarter note (because four quarters is one whole measure).
Under normal circumstances, an eighth note would be one half of a beat. You count it like Lawrence Welk: “One-anna-two-anna-three-anna-four-anna”. But, if your time signature is something-over-eight, then each eighth note gets a beat.
The most common time signature is 4/4, and it’s often called “common time”. Very straight forward. Pretty much all marches are in 4/4 (they have to be for the left-right-left-right part of the equation). On sheet music, common time is often denoted as a C. “This Little Light of Mine” is in common time (see below).
Next most common is probably 2/2 (that is, two beats per measure, and each beat is a half note), and it’s called “cut time” or “half time”, because you’re basically taking 4/4 and making it twice as fast. On sheet music, cut time is often denoted as a C with a vertical line through it, like a large cent mark.
Three-quarters’ time is 3/4 - three beats with each beat being a quarter note. Very simple to count: one-and-two-and-three-and. This is what waltzes and many other dances are written to.
Five-eighths time is counted like this: “one-two, one-two-three”. My first high school band director taught us to say “I love POST Toasties. I love POST Toasties.” to get the count right.
Seven-eighths time is counted like this: “one-two, one-two, one-two-three.” Both 5/8ths and 7/8ths tend to have fast meters.
There are some other really squirrely time signatures out there, but those are probably the most common. Just remember that, other than marches, the emphasis of the beat always falls on 2 and 4 in common time. Friends don’t let friends clap on one and three.
For example, when you sing “This Little Light of Mine”, you clap on the capitalized syllables, because that’s where beats two and four fall:
This LITtle light OF mine,
I’m GONna let IT shine.
This LITtle light OF mine,
I’m GONna let IT shine.
Hope that helps.