This overview will pretty much stay with black and white printing papers. There is a fairly wide variety of them, despite the fact that Kodak is quitting manufacture of theirs, and Forte's disaster. B&H carries six different brands of black and white enlarging paper, Freestyle nine. So there is still quite a variety available. The major distinctions of paper characteristics can be split between the physical and the photographic.
Quite simply, the size of the paper. Most papers come in 5 × 7", 8 × 10", 11 × 14", 16 × 20", and 20 × 24", with availability of the larger sizes a bit problematic. In recent years the off-the shelf availability is extremely limited, unless you happen to live in New York or Los Angeles, but most sizes and brands are pretty available to order. Also available are rolls in widths around 42", for really large printing.
Paper Type (Stock)
This is a major division. The most common type for general use is RC, or resin coated. The paper stock is coated with a resin that prevents absorption of liquids, in particular chemistry. This means that the paper can be washed thoroughly much faster than its counterpart, fiber paper. RC will also lay flat easier. The trade-off is that RC paper really has no character whatsoever. By comparison to decent grade of fiber paper, RC prints lack a certain depth, visually. I think this has a lot to do with the paper stock, and less to do with differences in the emulsion.
Fiber paper does not have the resin coating. This means that it takes longer to wash (see fiber processing). Not all, but many fiber papers available today are near or full exhibition grade. This generally means they have a thicker emulsion, and a heavier paper base. Although fiber takes more time and effort, it can make quite a difference in how an image looks.
This is the thickness or weight of the paper stock. Some light, most medium, and some heavy-weight. The heavy is often limited to exhibition-grade papers, and the light is getting a bit harder to find. As a rule, resin-coated (RC) papers come in a medium weight, with a medium-weight fiber paper being a bit heavier. Below is a comparison to common weights of watercolor paper.
Surface or Texture
This means the texture of the paper, rather than the surface sheen of the emulsion. Most papers are smooth, but a pretty wide variety of textures are available, such as linen, silk, or watercolor paper. These are more specialty items than standard fare. Much of this market is being moved to digital papers.
Referring to the tone of the paper itself, papers can range from a cool, blue-white, to a warm brownish white. Most are in the middle, and many are sold as cool tone or warm tone.
Probably the single most important photographic characteristic is the contrast of the paper. Here we can split into two camps. Traditionally, papers came as graded, or in standard contrast grades, from 1, which is flat, to 5, which is very contrasty, with 2 or 3 being standard. So, to make an image more or less contrasty you would need to change papers. There are still plenty of graded papers on the market.
Probably most used is variable contrast or polycontrast or multigrade. Variable contrast allows you to change the contrast of the paper by altering the color of light coming from the enlarger, most often by using polycontrast filters. You can get a range from very flat to very contrasty from the same box of paper.
This refers to the emulsion tone, wich, like the paper tone, can range from a cool blue-black, referred to as cold tone to a warm sepia, or warm tone. Developers can enhance the tonal characteristics of an emulsion, as well as toners.
Surface is the emulsion sheen, usually in three flavors: matte, pearl, and glossy. Glossy will give you richer blacks, but more reflections. Matte makes it very difficult to get a rich black. Pearl or semi-gloss is usually a good compromise. Glossy fiber dries to a sheen that is less reflective than glossy RC. Matte is generally only used for hand-coloring or other tangents where a tooth on the surface is desired.
Most black and white papers are about the same, normal, speed, which means they are designed for a reasonable exposure, which is measured in seconds rather than either fractions of a second or minutes. This is really because of the enlarging process. To get an idea of what normal is, a 35mm negative with a normal contrast range and correct exposure, enlarged to 8 × 10, should take around 8-10 seconds at ƒ/8. Around.