Most shade-buying mistakes come down to one number, the shade fitter size, and it isn't the one people measure. A shade fits by its fitter, the diameter of the opening where it attaches to the fixture, not by its overall width or height. Get the fitter size right and the shade mounts; get it wrong and a beautiful shade won't go on the fixture at all.
The fitter is the mouth of the shade. On most fixtures, the shade's opening slides into a holder or ring on the fixture, and the diameter of that opening has to match the holder. A few standard fitter sizes cover most old and new fixtures, two and a quarter inches being the most common, with others like three and a quarter and four inches showing up on larger fixtures. This is the number to know because it's the one that determines fit. A shade's overall size affects how it looks, but a shade whose fitter doesn't match won't mount no matter how good it looks, and two shades of very different width can share the same fitter. Measuring it is simple: measure the diameter of the shade's opening, the part that attaches, across the inside, and that's your fitter size.
Salvager's Note — how to measure a fitter: Measure the diameter of the shade's opening (the part that attaches), straight across the inside. That number, not the shade's overall width, is what has to match your fixture's holder. Common sizes: 2¼, 3¼, and 4 inches.