Assuming you're not talking about _Sphagnum_, which is all wet and
spongy and nice, I don't know what the moss is technically called. I'd
bet that moss taxonomy is really nasty, and you'd need a good microscope
to do it (I recall that to key lichens, you need a small arsenal of
chemicals to look for various reactions). I just call them all Bryophytes
and then pretend to know what I'm talking about!
It's not live peat moss. Peat moss is usually decomposed _Sphagnum_, or
sometimes decomposed sedges (a grasslike plant). Mostly in the US it's
from _Sphagnum_.
One kind of moss consists of zillions (a technical term) of little filaments
that rise out of the soil, like hairs. Michael Chamberland on this list
calls those "nose-hair moss" which I think is as good a name as any.
B