There are several ways of dividing a plant. One popular and very easy method is by brute force, simply dig up the plant, cut a chunk off it with a sharp spade and then replant both pieces separately.
This is very effective for some of the larger clump forming plants. It does leave a lot of severed material, which can rot and, in theory, cause problems, but which rarely seems to.
A refinement of this is to insert two forks into the dug up clump, back to back, and then lever them apart, forcing the plant into two pieces, continuing until the pieces are small enough. With the exception of the older, center pieces of the plant, which should be discarded, the pieces of plant can be replanted immediately.
It is, however, always a good idea to use the opportunity to rejuvenate the soil by digging it and adding some well rotted organic material. The best time to divide plants in this way is the spring.
A more sophisticated approach is to use your hands. Many plants if shaken so the soil falls off and gently manipulated with the fingers, will fall apart into separate crowns, each of which can be either replanted in the soil or put pots until they are needed.
This is only possible with plants grown in lighter soils, although it is still possible on heavier soils if the plant is manipulated in a bucket of water. More stubborn plants, such as hellebores, can have the water washed off first with a garden hose.
It is difficult to visualize what actually happens until you try it, and then you will be surprised how easily the plant comes apart in your hands. Some plants, such as hostas for example, may need a sharp knife to separate the crowns, but generally they just fall apart.
Normally, divisions are made in spring, but if the plants are transferred to pots they can be divided at any point during the growing season.