The term stay refers to standing rigging on a boat that runs either fore or aft of a mast. Stays are similar to shrouds in that they are usually steel cables that help hold up the mast, except that shrouds are on the sides of the mast. The luff of the jib or genoa foresail is shaped by being attached to the forestay.
Boats with a single mast, such as a sloop, usually have a single forestay and a single backstay. A cutter rig, however, also has an inner forestay to accommodate a second, smaller, jib sail.
It is common for sloops longer than 20 feet to have a jib roller furling system. In this case, the forestay is not a steel cable, but is an extruded aluminum band with one or more slots for the jib or genoa to slide into. Or, the extruded aluminum band (also called a foil) is shaped around the steel cable forestay, and the cable bears the force of holding up the mast, while the foil holds the luff and keeps it straight, and allows the foil to turn to furl the sail.
The backstay is usually a steel cable. This cable is usually fixed in place and tensioned with a turnbuckle. Some boats have a system for changing the tension on the backstay while under sail. Increasing the tension on the backstay creates more bend in the mast, which flattens the mainsail, which may be desired in different wind conditions; this flattening action is most effective in a fractional rig, in which the forestay attaches to the mast near, but not at, the top of the mast. Tighting the backstay on a masthead rig boat mostly tightens the forestay by tipping the mast aft, rather than bending the mast.
Sometimes shrouds are called sidestays.