Half mast is the term describing flying a flag below the summit of the flagpole (mast). This is done in many countries as a symbol of respect, mourning, or distress.
The tradition of flying the flag at half-mast began in the 17th century, perhaps to allow "the invisible flag of death" to fly at the top of the mast—which signified death's presence, power, and prominence. In some countries, for example the UK, and especially in military contexts, a "half-mast" flag is still flown exactly one flag's height down from its normal position, and no lower, to allow for this flag of death. This was the original flag etiquette. It is now standard, especially outside the UK, to fly the flag at halfway down the mast regardless of the size of the flag or hoist. (For modern UK practice see below.)
When hoisting a flag that is to be displayed at half-mast, it should be hoisted to the finial for an instant, then lowered to half-mast. Likewise when it is lowered at the end of the day, it is to be hoisted to the finial for an instant, and then lowered.
In the US, the term "half-mast" is commonly used colloquially to refer to half-staff, although US law and military tradition indicate that "half-mast" is generally reserved to usage aboard a ship, where flags are typically flown from masts. Not all English-speaking nations observe this distinction.