Qualifications
Aside from a high school diploma, the main qualifications for being an upholsterer are physical: manual dexterity and good eye-hand coordination are key. You'll be performing a variety of detail-oriented tasks, such as hand-stitching, measuring, working with a sewing machine, and hammering in nails and tacks.
You should also be able to stand for long periods of time—shifts are often eight to nine hours. You'll gain a new appreciation for quality (or, as your parents call it, "sensible") footwear and save those Toms for when you go out with friends. In addition, you don't have to be especially strong, but you'll definitely need to be able to move furniture around.
Now that we've covered the bare minimum, let's look at qualifications a more experienced upholsterer brings to the table. Many employers ask for experience with a variety of furniture, such as sofas, loveseats, chairs, and sectionals. You'll also be expected to be skilled in a variety of fabrics and other material, such as leather and vinyl. Tufting skills are totally a plus.
Perhaps most importantly, though, you'll need a good eye for detail. No one wants to see crooked stitching on their couch cushions, do they? No. No they don't. If you don't care about the tiniest, seemingly insignificant details, you probably won't enjoy this career.