Most freshwater fly anglers only occasionally, if at all, see their backing. It is still necessary to fill up the spool, which will give the fly line a larger diameter platform to rest on, and faster retrieval. Beware of having too much backing – it can lead to the line jamming against the non-moving part of the reel.
For freshwater, Dacron backing (woven synthetic) is excellent. A twenty-pound test is quite sufficient.
For saltwater backing, Dacron is good, but we might want to have 30–pound test strength.
Some backing is colored, and I have noticed occasionally that the color can bleed into the fly line, which does not look good to me. Therefore, I use white backing. However, colored backing is supposedly easier to see when the happy day arrives when you hook a monster fish.
A case can be made for Gelspun backing (synthetic, very strong, and abrasion-resistant) since it is much thinner than Dacron for the same test weight. That way you can put far more backing onto the spool. It will also cut through the water much better than the thicker Dacron. But beware, it can cut your fingers to the bone, so I do not recommend it for beginners.