To countersink a screw is to drill the screw so that the head of the screw is flush with the surface it is being drilled into. To countersink a screw, you must drill a pilot hole. Then, using a counter sink tool, drill a second hole wider than the first. Stop when the hole is as wide as the head of the screw. If you need to sink the head of the screw below the surface, the second hole should be drilled so that it is slightly wider than the head of the screw.
Countersink bits look similar to standard drill bits, except that they are wider and shorter. Countersink bits come in different sizes for different size screw heads. Countersink bits can attach to countersink drills, or they can attach to countersink holders.
Countersink holders and countersink handles are meant to attach to the countersink bits. These holders are turned manually to create the hole for countersinking the screw. Countersink holder/handle combos are typically less expensive than countersink drills.
Countersink drills are specifically designed to hold countersink drill bits. If you would like to use a power tool for your countersinking, but do not wish to purchase a drill specifically for countersinking, you can purchase a specialized countersink drill bit that works with a standard drill. These bits are more costly than standard bits.
Some screws are designed to countersink all on their own. Construction screws have special tips and heads that bite into the surface where they are being drilled. These types of screws will drill into any surface relatively easily, and without a pilot hole or countersink hole.