The main natural fibers are cotton, linen, silk, wool. We can also cite hemp or cashmere. They are therefore of plant (like cotton) or animal (like wool) origin.
Natural fibers are not necessarily ecological. Cotton, for example, is one of the most polluting products in the world and the most consuming of water. Flax and hemp are among the most environmentally friendly textile fibers in the world.
Natural fibers are biodegradable. They are easily degraded in the environment, unlike most synthetic fibers (although new processes exist to make synthetic fibers biodegradable).
The natural fibers are pleasant in contact with the skin. They are generally softer and more pleasant than synthetic fibers.
Technicality for sport: merino wool
For sports, the most technically interesting natural fiber is merino wool.
Merino wool is soft, does not itch, contrary to popular belief about wool in general.
Merino wool is a thermal regulator. It is an "active" fiber capable of regulating heat according to the outside temperature. It retains heat when it is cold and releases heat when it is hot.
Merino wool is anti-odor. You can wear a t-shirt or socks made from merino wool during exercise without worrying about the odors. Even for several days.
Merino wool is breathable (moisture management). It can absorb a lot of sweat before it gets really wet. Even when wet, it retains its thermal regulation function and avoids the icy effect of perspiration (unlike cotton).