Certain types of paper are made specifically for markers. These kinds of paper are typically thin and smooth, with a coating that prevents ink from bleeding through to the other side of the paper. That coating means that there is very little absorption, so marker paper uses less ink and takes fewer strokes than absorbent paper, such as card stock. As a result, trying to use many layers of coloring often won't work well.
Although glossy paper and vellum are different paper products, they share some similarities. Both are very smooth and nonabsorbent; the lack of absorbency means you use less ink, and in many cases, this results in colors being lighter. It is more difficult to blend colors on these kinds of paper. When working with glossy paper, you'll find that it's extremely difficult to change something you've already put down on the paper. You also don't have to worry about "feathering" -- ink spreading out across the paper -- so lines will be very sharp. Because vellum is transparent, you can blend colors by using a color on one side of the paper and a different color on the opposite side.
Card stock is somewhat more versatile than other kinds of paper. It is absorbent, and the thicker the paper is, the more absorbent it will be -- and the more feathering you will see on your artwork. Card stock for Copic markers is usually thin and smooth with tight fibers. Use this kind of paper to blend and layer colors and to apply advanced coloring techniques. Copic markers will bleed through to the backs of most kinds of card stock, which may be bad or good depending on your preferences.
Bristol board may work well for illustrators who prefer paper that is heavy and smooth. Thick fine art paper, such as acrylic paper, can result in brighter colors than other papers, but are too absorbent for detailed work. You can use Copic markers on a variety of mixed media surfaces; however, it isn't recommended that you use Copics on fabric that is meant to be worn. Also, you can use cheap paper, such as ordinary printer paper, if you're able to draw while taking into consideration how much it will bleed and feather.