A mortise is a rectangular hole designed to receive a tenon of the same size. Traditionally, mortises were cut by hand with a chisel. There is now a specialty machine called a mortiser that is used for cutting mortises. A mortiser has a square chisel with a drill bit inside that drills the hole and cuts the square edges of the mortise.
Sure, some drill presses have a mortising attachment that can turn your drill press into a hollow-chisel mortiser, but one really isn't necessary. If you don't have a mortiser (or a mortising attachment), you can use your drill press to cut mortises, and then square up the mortise with a sharp chisel.
Mark the outlines of your mortise in the wood with a pencil. Then, insert a drill but that is close to the width of the mortise into the chuck of the drill press. Adjust the depth of plunge on the drill press to match the depth of the mortise, then position one end of the mortise directly beneath the bit. Start the drill press and plunge the bit into the wood, taking care that the bit does not cut beyond the outlines of the pencil marks denoting the edges of the mortise.
Raise the bit out of the wood, then slide the board the width of the drill bit, and make another hole in-between the outlines of the mortise. Continue this process until most of the material has been hollowed out of the mortise.
Next, clamp the board in a woodworking vise, or in the absence of a wood vise, merely secure it on a workbench with a couple of woodworking clamps. Then, place the flat side of a sharp bevel-edged chisel against one of the pencil marks, and remove the remaining material to complete the mortise.