Short answer: clean, dry, and free of names on the label. That's the difference between clothing a shop can sell immediately and clothing that goes straight to recycling.
Underwear, swimwear, and items with heavy pilling or bobbling are usually turned away regardless of condition, for hygiene and presentation reasons. When in doubt, ask the shop directly — most would rather answer a quick question than receive something they have to bin.
Use the directory to check which shops near you currently need clothing before you make the trip.