When choosing paper, you need to consider that watercolors, acrylics, or pencils, for example, all have papers ideally suited to them.
Papers to use for drawing and sketching
Drawing paper is the obvious choice for sketching with pencils and is usually supplied in a pad or sketchbook.
Bristol paper (or Bristol board) has a really smooth surface and is a preference for some artists, but I find it too smooth. Unlike illustration board, both sides can be used.
You can also use hot-pressed watercolor paper for drawing, which has a smooth surface. My favourite watercolor paper for drawing on is the hot-pressed watercolor paper: Fabriano Artistico Watercolor Paper. Make sure you order hot-pressed (ie smooth) paper for drawing purposes. Cold-pressed paper has a rough surface and is suitable for wet watercolor washes.
Layout paper is a lot thinner than drawing paper and almost translucent. It is good for refining drawings as an alternative to repeatedly erasing one drawing. Lay the paper on top of another drawing to refine it.
Tracing paper is useful for transferring an initial drawing to another surface and for tracing from reference materials such as photographs.
Pastel paper is made especially for pastel drawings as it has enough texture to hold the pigment in place, although you can experiment with other materials.
Blotting paper is very absorbent and useful for placing under your drawing hand to avoid grease and marks getting onto the finished drawing.
Papers to use with watercolors
Watercolor paper is made in different weights, or thicknesses, and the lighter ones need to be stretched before using them or they buckle. Alternatively, you can buy pads that are glued at the edges to keep the paper flat during use. Lighter papers come in weights such as 140lb and thicker ones come in weights such as 300lb. I prefer thicker papers as they are easier to work with. They are more expensive, however.
Watercolor paper also has different textures. Cold-pressed watercolor paper has a rough surface suitable to holding very wet watercolor washes. Hot-pressed watercolor paper has a smooth surface and is suitable for more detailed work, such as botanical illustration.
Papers for use with oil paints and acrylics
You can buy papers for use with oil paints or acrylic paints. These are useful for creating preliminary work and working on ideas before starting work on a canvas.