9/8 is a
compound triple meter. You can think of it in 3/4 if each group of three eighths are thought of as triplets. Notation wise it's very different, but you could write something in 3/4 and 9/8 and it would sound exactly the same. This is how compound meters where created. Their sole purpose, in the beginning, was to give the composer a way to notate without having to write triplets.
Look ...
The quarter note gets the beat or pulse in 3/4. With 9/8, even though the eighth-note get's the beat ... we'll allow the pulse to be the dotted-quarter. Doing this, you could play each measure back to back and they would sound exactly the same. The only thing that has to happen is you go from the quarter-note being the pulse to the dotted-quarter-note being the pulse.
6/8 is
compound duple meter. It's normally felt in two, despite the fact that the eighth-note gets the beat. You could write two groups of eighth-note triplets in 2/4, and it would sound like 6/8 ... as long as you do what we did earlier ... change from the quarter-note pulse to dotted-quarter-note pulse. However, look at this next notation:
Here you can see how even 3/4 can sound like 6/8. In this case, you don't not need to change the pulse; both have the quarter-note as their pulse.