Jibs and spinnakers are the two main types of headsails on a modern yacht.
Confusingly, any jib is technically a type of staysail, but the inner jib of a yacht with two jibs is called the staysail, and the outer (foremost) the jib. This combination of two jibs is called a cutter rig or a yankee pair. A yacht with one mast rigged with two jibs and a mainsail is called a cutter. A gaff-rigged cutter may optionally also carry a gaff topsail.
A fully rigged schooner has three jibs. The foremost one sets on the topmast forestay and is called the jib topsail, a second on the main forestay is called the jib, and the innermost is called the staysail.
On yachts with only one jib, it is common for the tack of the jib to be further aft than the mast, meaning the jib and mainsail overlap. An overlapping jib is called a genoa jib or simply a genoa.
On cruising yachts, it is common for the innermost jib to be self-tacking, either by using a boom along the foot of the sail or by cleating the sheet to a track. All other jibs need to be worked when tacking, that is, as the yacht comes head to wind the sheet attached to the tack of the sail is released, and then pulled in again on the other side of the yacht.
See also: sail-plan