A breathable cover is a cover or bag made of a material that lets air and water vapour pass through, such as cotton, muslin or a non-woven fabric like Tyvek, so it keeps dust and light off an item without trapping moisture against it. It's the alternative to a plastic cover, which blocks air and holds in any damp underneath.
At Wigwam
The covers in our Box Shop are polythene, not breathable: a 2 to 3 seater sofa cover (£4.99), a pack of two chair covers (£4.99), single and double to super king mattress covers (£3.99 and £5.99) and a 3m x 5m dust cover (£5.49). They're made to protect against dust, dirt, moisture and marks while you move and store. Our own guide to climate control recommends breathable wrap for wood, and "wrapping wooden surfaces in breathable protective material" for veneered and inlaid pieces, so for those, bring a cotton cover of your own.
Which to use
A breathable cover suits anything that holds moisture or suffers from it: upholstery, wooden and veneered furniture, good clothes, quilts and rugs. The Canadian Conservation Institute advises against plastic garment bags for textiles "because there is no air circulation (which can encourage moisture to build up)". Museums Galleries Scotland suggests washing cotton dust covers before use to remove impurities.
Put simply, our polythene covers are for the move itself, over clean, hard, dry items where drips and dust are the risk, and they come off once the item is in the unit. For upholstery, wood or fabric in storage, and certainly over a winter, cloth is the cover to leave on. For leather, the Canadian Conservation Institute advises wrapping or covering it in humidity-buffering materials such as cotton or acid-free tissue; clean and dry it first. Whatever you choose, the item must be clean and completely dry before it's covered, because no cover dries out damp that's already inside.
Related terms: Dust sheet · Sofa cover · Mattress bag · Ventilation
Related questions: How do I stop clothes going musty in a storage unit? · How should I store a mattress in a storage unit?