To get a screwhead flush with the wood, you must countersink it in a shallow, tapered hole. Drill a pilot hole with a straight bit (in hardwood, drill a hole for the screw’s shank too; see Drilling). Then center a countersink bit in the pilot hole and drill. The countersink bit should be the diameter of your screwhead, but a slightly larger bit will do if you stop short of its full diameter. Test for fit by inserting the inverted screwhead.
A faster method is to drill the pilot and countersink holes simultaneously with a combination bit specific to one size and length of screw. Unless you use the same screw size many times, this is an expensive method. To counterbore (sink the screw below the wood’s surface), use a bit exactly matched to your screwhead.