A square yard is exactly nine square feet, because a yard is three feet and three squared is nine. Divide by nine to go one way, multiply by nine to go back.
The conversion survives because of carpet. Carpet and vinyl are still quoted per square yard in much of the United States, while rooms are measured in feet, so a 300 square foot room is 33.3 square yards of carpet before waste. A price that looks high per square yard is often the same price as a rival’s per square foot — the units are doing the marketing.
Carpet is also sold from rolls of a fixed width, usually 12 feet, which is why a fitter’s quote covers more square yards than your room contains. Convert your measurement first, then ask how much roll the job actually needs.