It is great to hear that you are interested in the environment. Most napkins will end up in the trash when people are done with them. They are difficult to recycle because (1) they are dirty and (2) the individual bits of paper in the napkin are very thin and short. It is possible to wash and recycle a piece of cardboard or newspaper, but every time paper is recycled it gets thinner and shorter.

Once the napkins are in the trash, they can go to different places. Most of the trash in the US is put into landfills. In these landfills trash is buried in the ground and left to be decomposed (very very slowly) by microbes. A subset of the trash is burned in so called Waste-to-energy plants and the energy from the fire can be used to make electricity.

If you want to produce less trash, one option is to use napkins and towels made of cloth instead of paper. Ideally, you can use these clothes several times before washing them together with your laundry.