An ax is a handy tool to have around the garden -- you can use it to quickly strip large branches of smaller twigs, and there's nothing better when you need to split logs for firewood, remove stumps and make kindling. To get the most from your ax, you have to keep the blade sharp. As long as you don't want to do any carving, you only need a utility edge, which is sharp, but not sharp enough to shave slices from wood. Sharpening the ax is a job to do by hand, wearing protective gloves; an electric grinder can ruin the blade.
Clamp the ax firmly in a solid bench vise with the blade facing up. Put on cut-resistant leather carving gloves, because one false stroke during the sharpening process could result in a deep wound.
Hold a coarse file at about a 10-degree angle with respect to the blade -- it should be almost parallel -- and stroke down on the blade repeatedly until all the nicks and gouges on that side are gone and you can see the metal shining.
Move the file over onto the other side of the blade and repeat the procedure. Stop filing when the edge of the blade forms a burr, which is a paper-thin layer of metal.
Replace the file with a finer one, and repeat the procedure to remove the scratches and nicks left by the coarser file. At this point, the blade is very sharp, so keep your hands as far away from it as possible.
Finish the edge by holding a sharpening stone on one corner, on one side, of the blade and drawing it in a smooth motion along the edge to the other corner. Repeat on the other side of the blade. Continue in this way, alternating from one side of the blade to the other, until you're satisfied with the edge.