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The projecting piece (a tenon) on one board is inserted into the cavity (a mortise) on another, then glued. The joint may be further secured with a peg inserted through both pieces, like a toothpick in a club sandwich. Often used in armoires and cabinets.
Wedge-shaped projections on one piece interlock with corresponding slots on another. Of roughly the same quality as a mortise-and-tenon joint, this is typically used in drawer construction.
Small wooden rods (called dowel rods) are inserted into holes cut into connecting pieces of wood. Glue helps to secure the joint, but it's only as strong as the dowel rods are. Typically found in assemble-at-home furniture, such as Ikea bookshelves.
The boards are either glued and screwed together or fastened with nails. Typically the fastener goes through the face of the board that overlaps the butt end of the other.