Electrospinning is a process that creates nanofibers through an electrically charged jet of polymer solution or melt. This technique is applicable to virtually every soluble or fusible polymer and is capable of spinning fibers in a variety of shapes and sizes with a wide range of properties to be used in a broad range of biomedical and industrial applications. Electrospinning requires a very simple and economical setup but is an intricate process that depends on several molecular, processing, and technical parameters. This article reviews information on the three stages of the electrospinning process (i.e., jet initiation, elongation, and solidification). Some of the unique properties of the electrospun structures have also been highlighted. This article also illustrates some recent innovations to modify the electrospinning process. The use of electrospun scaffolds in the field of tissue engineering and regenerative medicine has also been described.