One yard is exactly 36 inches. Both units are defined from the same 1959 agreement, and the relationship between them is exact in both directions with no rounding anywhere.
About the units
The yard has been exactly 0.9144 metres since the International Yard and Pound Agreement, and the inch exactly 25.4 millimetres — and 36 × 25.4 = 914.4, which is the same number, confirming that the two definitions are consistent. The yard's traditional role is as a cloth measure, and the inch's is as its subdivision. Before decimal fractions were common, cloth was sold in binary subdivisions of the yard: a half, a quarter, an eighth and a sixteenth of a yard, the last of which is 2.25 inches. Traces of that survive in fabric shops, where a quarter yard is a standard cut and equals nine inches.
The exact factor
The factor is exactly 36, which is 3 × 12 — three feet to the yard, twelve inches to the foot. Every step in the chain is exact, so any conversion in this pair can be done without approximation. Useful reference values: a quarter yard is 9 inches, a third of a yard is 12 inches, a half yard is 18 inches. Because 36 has many divisors, most practical fractions of a yard land on a whole number of inches, which is exactly why the duodecimal structure was kept.
Where you meet this conversion
Fabric, ribbon and trim are the dominant case. American quilting and dressmaking patterns specify yardage while the cutting instructions are in inches, and a pattern that calls for two and a half yards needs 90 inches of usable width. Upholstery and curtain making work the same way. In sport, American football gives yardage while equipment and field markings are in inches. Rope, chain and webbing are often sold by the yard with diameters in inches, so a single order mixes both units on one line.