Countersinking is to enlarge and/or bevel the side of a hole to allow any given fixing to be sunk below the surface of that hole. When the fixing has been put in, the countersink can then be filled and sanded to allow the surface to be completely flat with no visible signs of fixing.
A pilot hole is a preparatory hole drilled to allow easier passage of a larger object. In this case a pilot hole is drilled to allow a screw to be driven in making its passage through the timber easier. Because the pilot hole has a diameter of less than the screw itself, the thread of the screw can still bite into the surrounding timber making a perfectly secure fixing.
Some timbers, especially when cut into narrow or thin strips, are prone to splitting when nailed or screwed. Pilot hole and countersinking prevent this happening. If the fixing, even a nail, does not have so much timber to displace as you force it through the timber, it is much less likely to split. And if, because of a countersink, it can slip down below the surface of the timber, there is little chance of it splitting as the conical shape of the underside of the screw head touches the top of the timber.
Measure the diameter of the screw, for example a number 8 screw has a diameter of 3mm, from outside of thread to outside of thread. A sensible size for the pilot hole is therefore 1 or 1.5mm. This will still allow the thread to cut its way through the timber without splitting it, while getting great grip on the timber. A bradawl or gimlet can be used to mark the point to place your small drill.
Pilot holes are used almost all the time through metal as it is impossible for most screws, even run of the mill self tapping screws, to force their way through. Countersinking is used also to get the head of the screw flush with, or below, the surface.
Countersink bits are available for this job. It is an excellent idea, when fixing to a concrete lintel for example, to drill a series of pilot holes, one through the other and getting slightly larger every time, until the required hole size is met. Trying to drill a very large hole through both concrete, hard masonry and metal, will quickly blunt the drill as cutting (drilling) with a large surface area is very much harder than with a small one. Place the countersink bit in your drill and push down lightly on the pilot hole. For those who prefer a hands on approach, a hand held countersink tool can be bought by clicking on the tool box below.