A hydrofoil is, as its name implies, a type of foil (or wing) intended for use in water. That is, it is an object which, when placed in a water flow, generates lift perpendicular to the direction of the flow across its surfaces.
As airfoils are used on aircraft, so hydrofoils are generally used on watercraft. The most common use of a hydrofoil is to lift the hull of a boat out of the water at speed. As the hull is lifted further up in the water, the drag on the boat drops dramatically, and the craft can increase its speed. High-performance hydrofoils will manage to lift the entire hull out of the water, with the foils themselves mounted on projections below the hull itself.
The term 'hydrofoil' is generally used to refer to a watercraft that utilizes hydrofoil lift. Although they are quite fast, hydrofoils can be difficult and dangerous to use if the sea state is too rough, as waves high enough to reach the hull at speed can cause significant shocks or even destabilize the boat.
Hydrofoils on boats are generally divided into two types - the surface-piercing hydrofoil, which is usually mounted at an angle roughly parallel to the bottom of the hull, and thus rises out of the water at the outboard edge as the lift mounts, and the fully-submerged hydrofoil which is mounted horizontally below the hull on structural extensions. These offer better lift, as they don't lose effective area as the craft rises; however, the surface-piercing hydrofoil offers better stability as the lift vectors angle in towards the boat's center of gravity, and require less clearance beneath.