Rag ~ paper made from all cotton fibres. They are stronger and better suited to artists' papers. I am not sure whether linen fibres are included in this definition ~ they are even stronger and often used in a blend.
Bright ~ a term sometimes used to describe bleached papers. There are some chemical additives that increased perceived brightness ~ laundrymaids added a little blue dye to increase the 'brilliance' of their whites. 'White' cotton materials and papers with impurities tend to yellow in time and the blue helps to counteract this. Residual bleach, or other additives, may affect pigments as well as the paper fibres. Wood pulp is more prone to this ~ hence artists' preference for rag papers that are acid-free.
Acid~free papers are those made without the use of acids ~ which remain in some papers in minute quantities even if well washed. The acids ~ over time ~ will deteriorate the fibres and also some pigments. Handmade papers are typically made only with the use of pure water ~ a feature which some sellers use in their marketing materials.
Hot-pressed paper is ~ in laundry terminology ~ ironed to give a smooth surface. Many artists prefer a slightly rougher surface ~ see NOT. Printers ~ who lay inks down on the paper ~ prefer not to have bumps and valleys in the surface which lead to irregular quantities of ink. The smoothness of handmade papers is determined the by roughness of the blankets used during coucinigand subsequently. Our Nepalese Hatakami paper is air dried on metal plates to give a smooth surface.
NOT ~ always capitalized ~ is neither rough nor smooth but comnsidered 'just right' for watercolourists ~ for whom 'sizing' also plays an important part.