The definition of a synthetic fiber is a "man-made" fiber.
Semisynthetics involve taking a natural fiber that is produced by plants and animals and improving upon it or changing it to make the naturally occurring fibers stronger and better.
True synthetics are also formed in a number of different ways through a number of different processes.
Nylon, which actually refers to a class of different polymers that are referred to under the scientific term polyamides, were the very synthetic fibers to achieve widespread use in the US.
An example of a synthetic fiber is Polyethylene terephthalate (PET) polymer, a true synthetic, which is produced from ethylene glycol and either dimethyl terephthalate (DMT) or terephthalic acid (TPA).
Examples of synthetic fibers are viscose rayon, acrylic and polyester.