Door jambs suffer almost as much wear as doors do -- people bump them and tight-fitting doors scrape off finish. The proper way to restore one is to sand it down to the bare wood, restain it and then spread a new finish. That much sanding is a lot of work, however, and it probably isn't necessary. In most cases, all you really need to do is fix dents and holes, hand-sand the jamb and then stain right over the old stain. A gel stain will soak into bare wood and spruce up the old finish before you clear-coat.
Tap out the hinge pins with a hammer and nail and take the door down. Unscrew and remove the hinges and door strikes from the jamb with a screwdriver.
Tape the edges of the jamb to prevent staining the wall. If you're including the door casing in your refinishing job, lay painter's tape on the wall along the edge of the casing. If you're not going to work on the casing, bring the tape to the edge where it meets the jamb.
Wipe the wood down with mineral spirits to assess its condition. The mineral spirits will highlight scratches and dings as well as small patches of paint. Note the areas that need work.
Scrape paint off with a razor blade, holding the edge of the blade parallel to the surface and moving the blade in a direction perpendicular to its edge. Scrape lightly so you remove paint, but not the finish underneath.
Fill holes and dents with epoxy wood filler. Use a product that is the same color as the stained wood, because you can't change its color with stain. Trowel it on with a putty knife; scrape off as much excess as you can, and then wipe off the rest with mineral spirits before the filler sets.
Sand the jamb by hand with 150-grit sandpaper. The purpose of sanding is to etch the finish, not to remove it. Sand with the grain of the wood. Fold the paper in thirds and work the edge into the corner between the stop and the jamb.
Soak a rag with gel stain that is the same color as the stain already on the jamb, and rub the jamb to work in the stain. Use strokes that are parallel to the grain. After you've stained all the wood, go back over it with a clean rag to wipe off excess stain.
Apply a coat of clear polyurethane finish with a paintbrush. Let the finish dry, and then sand the jamb lightly with 220-grit sandpaper and apply another coat. Let that coat dry, then replace the door hardware and rehang the door.