Covers: generic term for anything on a bed.
Blanket: a piece of relatively thick cloth that helps keep you warm. Traditional materials for a blanket are wool, cotton, etc.
Sheet: A thin piece of cloth that that is often fitted onto the bed. So, you lie on the sheet. There is often a 'top sheet' as well in the US, so you sleep on top of one sheet, and under another one. If the weather is hot, you might sleep just under a sheet, without a blanket. Traditionally, sheets are made of cotton.
Quilt: Same function as a blanket, but they are often thicker. Also, they are made (the outside at least) of many, possibly random pieces of cloth sewed together.