Electrospinning is a simple and versatile method to generate nanofibers. Remarkable progress has been made in the development of the electrospinning process. The production of nanofibers is affected by many parameters, which influence the final material properties. Electrospun fibers have a wide range of applications, such as energy storage devices and biomedical scaffolds. Among polymers chosen for biological scaffolds, such as PLA or collagen, polyacrylonitrile (PAN) has received increasing interest in recent years due to its excellent characteristics, such as spinnability, biocompatibility, and commercial viability, opening the way to new applications in the biotechnological field. This paper provides an overview of the electrospinning process of a large range of polymers of interest for biomedical applications, including PLA and PEO. It covers the main parameters and operation modes that affect nanofiber fabrication. Their biological applications are reviewed. A focus is placed on PAN fiber formation, functionalization, and application as scaffolds to allow cell growth. Overall, nanofiber scaffolds appear to be powerful tools in medical applications that need controlled cell culture.