In compound time
In all frequently used compound time signatures the bottom number is 8. Three 8th notes (quavers) make a beat.
Divide the top number by 3 for the number of beats to the bar. In 6 - 8 time there are six eighth notes to the bar, two compound beats to the bar.
Three quavers make a dotted crotchet.
By putting a dot to the right of a note head you increase the note's time value by half. So a dotted crotchet has the time value of one and a half crotchets, 3 quavers, one beat of compound time.
Usually (but not always) the compound time beat is a dotted crotchet.
|3 - 8 Compound time
1 dotted crotchet beat to the bar.
|6 -8 Compound time
2 dotted crotchet beats to the bar.
|9 - 8 Compound time
3 dotted crotchet beats to the bar.
|12 - 8 Compound time
4 dotted crotchet beats to the bar.
Is 9 - 8 always compound time?
9 - 8 says 9 quavers to the bar. It does not say how the quavers are grouped. Three dotted crotchet beats make 9 - 8 compound time. Three crotchet beats and one dotted crotchet beat makes 9 - 8 uneven time.
The time signature alone is not enough to show the feel. We need beams. We need groupings. We need to see the beat.
Why notate 3 - 8 and 6 - 8 with beamed quavers and 9 - 8 and 12 - 8 with dotted crotchets?
Well, it has to do with the feel. The time signature alone is not enough to give you the feel. You need the beams too.