The tool you need for cutting a round hole in wood depends on the size of the hole. Drill bits can cut holes up to 1 1/2 inches in diameter, while hole saws are best for circles up to 4 1/4 inches. Cut larger circles with a jigsaw.
Plumbers and electricians often turn to spade bits when they need to cut holes in framing for pipes or wire, but woodworkers prefer Forstner bits, which make cleaner holes. Like these bits, hole saws are also drill attachments, and they are useful when the hole diameter is between 1 1/2 and 4 1/2 inches. Carpenters commonly use hole saws to cut through doors when installing locksets. Hole saws develop considerable centrifugal force, which increases with diameter, so they aren't practical for holes larger than about 4 1/2 inches. You need a jigsaw for these.
To cut out a circle with a spade bit, Forstner bit or hole saw, start the hole from one side of the piece of wood and finish it from the other. This prevents chipping. When cutting with a jigsaw, use a 1/2-inch drill bit to make a hole for the saw blade, ensuring that the outer edge of the hole just touches the line you scribed for the circle. Carefully guide the saw around the circle, keeping the outer edge of the blade on the line.