Natural cellulose fibers are fibers that are still recognizable as being from a part of the original plant because they are only processed as much as needed to clean the fibers for use. For example, cotton fibers look like the soft fluffy cotton balls that they come from. Linen fibers look like the strong fiberous strands of the flax plant. All "natural" fibers go through a process where they are separated from the parts of the plant that are not used for the end product, usually through harvesting, separating from chaff, scouring, etc.
Manufactured cellulose fibers come from plants that are processed into a pulp and then extruded in the same ways that synthetic fibers like polyester or nylon are made. Rayon or viscose is one of the most common "manufactured" cellulose fibers, and it can be made from wood pulp.
In the textile industry regenerated cellulose is used as fibers such as rayon, (including modal, and the more recently developed Lyocell). Cellulose fibers are manufactured from dissolving pulp. Cellulose-based fibers are of two types, regenerated or pure cellulose such as from the cupro-ammonium process and modified cellulose such as the cellulose acetates.
The first artificial fiber, known as artificial silk, became known as viscose around 1894, and finally rayon in 1924. A similar product known as cellulose acetate was discovered in 1865. Rayon and acetate are both artificial fibers, but not truly synthetic, being made from wood. Although these artificial fibers were discovered in the mid-nineteenth century, successful modern manufacture began much later.
What is often marketed as "bamboo fiber" is actually not the fibers that grow in their natural form from the bamboo plants, but instead a highly processed bamboo pulp that is extruded as fibers. Although the process is not as environmentally friendly as "bamboo fiber" appears, planting & harvesting bamboo for fiber is much more sustainable and environmentally friendly than harvesting more slow growing trees and clearing existing forest habitats for timber plantations.