Nails are the most basic means of joining wood together and have been used in more or less the same way for thousands of years. Nowadays, nails are still used to penetrate even thick pieces of wood and join them together. Nails are preferred in situations where a joint doesn't need to stand up to a lot of pressure pulling in the direction that would uproot the nails, though the nails can take a lot of punishment from either side. Compared to screws, nails have the advantage of being quicker and easier to apply.
Screws are more commonly used than nails for joints that are likely to see a lot of stress. The spiral threads of the screw mean that the finished joint will hold up against a lot of pull from any direction. Screws need to be applied with a winding motion and are usually applied using a screw gun, especially when working with hardwood and in a project that will require many screws. It is difficult to install a screw without using a power tools, or without using something like an awl to drive a pilot hole.
Staples for wood are just like staples for paper, only much bigger and thicker. Even at that, they should be used for only light joining. Staples will generally be used alongside screws and nails for projects like attaching a large, thin sheet of wood to a thicker frame. The screws or nails provide the major strength of the joining, while the staples tack down the wood to prevent the wood's warping and lifting up away from the larger piece.
Wood glue is a milky, water-soluble glue similar in formula to white craft and school glue, only stronger. Wood glue is generally used for light, delicate woodwork, or as a means to reinforcing and strengthening another type of joint. Wood glue creates a strong seal that will stretch over a wide area and help prevent gaps from forming, but it's not strong enough to resist strong amounts of weight and force pulling against it. This makes it unsuitable as the sole joining method for any project larger than, say, a handheld item.
Wood biscuits are oval, semi-flat discs of reconstituted sawdust used to join two flat planes of wood together end to end; one such common application is for the creation of a tabletop. To use, woodworkers cut a slot into each ends of the wood and insert a biscuit, covered with wood glue, into one of the holes. When this process is repeated along all the length of the plank, it creates a tab-and-slot joint, which, when reinforced with plenty of glue, makes the two planks into a single, solid piece. The moisture from the glue will also cause the wood biscuits to swell, strengthening the joint.