The first thing I do with any cover is pinch the fabric and give it a tug, because that tells you more than reading the label ever will. Thin, slippery stuff will look fine on the hanger and then slide around on a cushion or bunch up under a duvet insert within a week. You want something with a bit of weight and grip to it, a cotton or a linen blend that pulls taut and stays put.
Covers live a hard life compared to most soft furnishings, since they're the layer that takes the coffee spills, the dog hair, the toddler elbows, so how they age matters more than how they look on day one. A good cover washes well for years if you treat it right from the start, cold water, mild detergent, and resist the dryer on anything with a weave you like the texture of, since heat flattens that out fast. Linen gets softer and a little more lived in with every wash, which I happen to like, though I know people who find that look sloppy rather than relaxed. Cotton holds its shape longer and shows less of that change. Either way, iron or steam while slightly damp if wrinkles bother you, because bone dry cotton fights you the whole way.
Sizing is where most of the disappointment happens, and it is almost always because someone measured the item, not the result they actually want. A chair cover needs you to measure the chair at its widest points, arms included, and check the drop length against the floor, not just the seat height. A sofa cover wants the same plus a few extra inches of give, since a cover pulled drum tight across cushions looks stretched and cheap, while one with a little ease drapes properly. For a duvet, match it to the insert size exactly, a queen cover on a full insert leaves you with sad, sagging corners. For pillow covers, go slightly smaller than the insert, maybe an inch, so the cover looks full rather than deflated.
On delivery day, check the closures first, zippers, ties, or envelope backs, since that is what fails first and is the most annoying thing to deal with once it's full of feathers. Check that seams are straight and the pattern lines up at any visible seams, and look at the underside too, not just the front, because that's where corners get cut. Give it a smell test too, some shipping treatments linger longer than you'd expect.
This kind of thing suits a home where the furniture gets used hard and the cover is doing the forgiving for you.