Different manufacturers may have trade names for each type of finish. Glossy produces a very shiny finish with good detail. Semi-matte softens the image and helps conceal grain. Matte has the dullest finish and the grainiest appearance and shows the least contrast. Match the image you want to produce to the type of paper.
Papers also come in various tones.
A cold tone paper has a distinct blue-black finish. Warm-tone papers have a creamier finish. The coldness or warmness of papers can be substantially altered by using different paper developers or through toning. Warm tone papers produce good portraits. Some require less exposure time than others to produce an image.