I used to sell carpet to builders so here goes.
Nylon = Looks great, has soft feel, reasonably hard wearing, will burn or melt a bit. Used in the majority of homes these days.
Wool = Looks great, has soft feel, very hard wearing, excellent heat insulation, wont melt, wont burn, pretty expensive. Not used in homes as much as it was due to price but is still the most desirable. Common in commercial buildings due to hard wearing and fire resistance.
Polypropalene = Looks pretty good, has rough feel, extremely hard wearing, melts easily {(dont get it if you smoke) a cigerette could easily melt a hole in it}, cheap price, commonly used in holiday homes or by people on a tight budget.
Marine or hospital grade = Looks vary (usually this carpet is very thin so that it can be hosed down on a boat or moped in a hospital) but you can get some unusual patterns, Extremely hard wearing, easy to clean and does not matter if it gets wet, it wont rot or get water marks as wool and nylon can.
Underlays = In homes and Commercial buildings rubber is almost always used. Most of it is made by Bridgestone. You can get different thicknesses and styles depending on what you want. Felt is now uncommon eccept for cars.
Auto carpet suppliers have plain felt (conpacted wool) or felt with a layer of bitumen in between for sound deadening. Felt is great for reducing heat but will mean that you will need to pull your carpet up if it gets wet. Then again you will probably have to do that anyway if you dont want the floor to rust out.
Some carpet places sell groovy patterns. I remember we used to have leopard and zebra patterns. Soooo many bedrooms were done with leopard skin pattern