Nanotechnology has attracted great attention from researchers in modern science because nanomaterials have innovative and superior physical, chemical, and biological properties, and they can be altered and modified accordingly. As particles get smaller, their surface area increases compared to their volume. Electrospinning is one of the advanced techniques to produce ultrathin nanofibers and membranes, and it is one of the best ways to create continuous nanomaterials with variable biological, chemical, and physical properties. The produced fibers can be utilized in various domains such as wound dressing, drug release, enzyme immobilization, etc. This review examines the biomedical nanofibers/membranes produced by electrospinning techniques to investigate the effects of process parameters (e.g., solution characteristics, applied voltage, and ambient conditions) on nanofiber characteristics (physical, chemical, and mechanical properties). The solution parameters like (i) optimum concentration, (ii) higher molecular weight, and (iii) higher conductivity produce uniform nanofibers, smoother nanofibers, and a smaller and more uniform fiber diameter, respectively. In addition, process parameters such as (i) higher voltage and (ii) slower flow rate produce more polymer ejection from the nozzle and enhance the smoother fiber production, respectively. The optimum tip-to-collector distance is considered to be 13–15 cm. The ambient conditions such as (i) higher humidity and (ii) higher temperature produce thicker and thinner nanofibers, respectively. The controlled parameters through optimization process determine the size and quality of the fibers. The effects of each parameter are discussed in this review. The applications of nanofibers are also discussed.