One mile is exactly 1,760 yards. Like the 5,280 feet it corresponds to, the number comes from the collision between the Roman mile and the English furlong, resolved by statute in 1593 in favour of keeping the furlong intact.
About the units
The yard is the defining unit of the imperial system, exactly 0.9144 metres since 1959. The mile is defined as eight furlongs, and a furlong is 220 yards — the length of a medieval ploughed furrow. Eight times 220 gives 1,760. Seen from the yard's side rather than the foot's, the structure is tidier than it first appears: 22 yards make a chain, 10 chains make a furlong, 8 furlongs make a mile. The 22-yard chain, devised by Edmund Gunter in 1620, was the instrument that made English land surveying practical, and it is why a cricket pitch is 22 yards long and why American city blocks so often measure in multiples of it.
The exact factor
The factor is exactly 1,760. The chain of relationships behind it is worth writing out because each step is exact: 1 mile = 8 furlongs = 80 chains = 1,760 yards = 5,280 feet = 63,360 inches. That last number is the one printers and map-makers used, since a map at a scale of one inch to the mile is 1:63,360 — a scale that appears on British Ordnance Survey maps for most of the twentieth century.
Where you meet this conversion
Golf is the everyday case, with course lengths given in yards and total distances occasionally in miles. Athletics kept the mile as a race distance long after metrication and reports splits in yards in older records; the four-minute mile was run on a track marked in yards. Land surveying and property descriptions in the United States and older British deeds use chains and furlongs, both of which resolve to yards. Military range estimation traditionally used yards for the same reason.