Nike and Adidas are brands. They do have designers working for them, but the designer's names aren't put on the products - the brand is the thing.
Some chain stores (Debenhams for example) sell clothes tagged with the designer's name (Jasper Conran, John Rocha for example) and those are designer clothes. They are usually fashionable with some additions to make them stand out a bit from the herd, may be of higher quality than major brands or non-designer labels, but are usually more expensive than their equivalent non-designer counterparts.