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I'm trying to build my first home-build computer.
(Asus A7N8X-E) And now I have a question about how
I do installing my MoBo in the case. I have fit
those copper things to keep the distants between the
case and the MoBo. On the MoBo side I have to fit
these little ring for the electrical insulation (or
is this isolation!!!)
do I fit those rings on both sides, or just on the
top side of the MoBo.

The copper things are actually brass and called "standoffs."

When in doubt, install those rings (fiber washers) on both sides of
each hole because if you leave out a needed washer, the motherboard
will short to ground and possibly be damaged. On the other hand if
you install unneeded washers you'll simply waste some time and washers
and have marginally worse immunity to electrical noise.

Nobody should make a blanket recommendation about insulator washers
and all motherboards, and each board must be inspected individually.
If a side of a mounting hole is surrounded by a copper donut (usually
has 6 tiny holes around it, may be coated with solder) or is
completely clear of any copper (neither standoff nor screw can
possibly touch copper traces), then it doesn't need an insulator
washer. Motherboards made in the past 5 years rarely need insulators
but I've seen exceptions as new as 12 months old.