Refinishing the piece means taking off the old finish and creating a new one. It's physically demanding, so be prepared to use some elbow grease for this messy and cumbersome job. For a do-it-yourselfer, it's best to use a combination of chemical strippers and sanding. Above all, be patient, even though some finishes can be stubborn to remove. Trying to speed up the process could result in even more repair work.
Research the type of stain you'll need for your wood. If it's not worth staining, you can always paint the surface. Just be sure it's not a valuable piece, because it won't be once you paint it. Before painting, sand the finish to smooth out any bumps. Regardless if you're staining or painting, you'll want to give the piece another good cleaning once you finish stripping and sanding to ensure that you end up with a professional-looking finish.