Hi. I'm new to inkscape and I'm making a background for my site. I want to remove where the smaller circles intersect the larger circles, as I tried to do on the picture below with the outer ring of the circle, so that it appears if the smaller circles are behind the larger circles (they have transparency). Any help would be appreciated, sorry for the confusing description.

Welcome aboard!

Here is a method producing the result I assumed you want to achieve:
Select all circles, convert outlines to paths by pressing Ctrl+Alt+C.
Select all the smaller ones, by clicking on each while holding Shift,
and press Ctrl+K to combine them,
do the same to the larger ones as well.
After that, select the larger circular path, duplicate it by pressing Ctrl+Shift+D,
and break it apart by pressing Ctrl+Shift+K,
then click on the larger circle while holding shift and delete the remaining selected circles.
After that, select the large circle and the smaller circular path,
and press Ctrl+-.
If you would like to have some space between them, scale up the large circle before the substracting.
That is, by setting proportional scaling at the top boxes, and using the corner handles of the bounding box, while holding down Shift.

Thank you! Is there a way I can remove the smaller circles from the space between the larger circles too?

Seems I couldn't explain it right or you jumped through some steps.
You combined the small circles together, after converted their outlines to paths.
Same with the larger circles.
Then, select the larger path -made up of the larger circles-, duplicate it, break it apart, and delete all the circles
EXCEPT the largest one that you use for subtracting from the smaller circles.
Maybe that was missed.