What filter should I use for slow shutter speeds on a bright day?

Filters modify the light that reaches the sensor. Depending on the type of filter they may allow only certain colors to pass (colored filters), reduce the amount of light reaching the sensor without regard to color (neutral density filters), allow only polarized light (polarizers), diffuse the light slightly (soft focus filters), cause certain kinds of highlights (e.g. starburst filters), etc.

If you want to decrease the shutter speed you're using (longer exposures) without reducing aperture or sensitivity (ISO), you can use a neutral density filter to block some of the light. Think of it as sunglasses for your lens -- it's a gray looking filter that simply absorbs some light. They come in different densities, and you can even get a gradient ND filter that'll block more light from one side of the image than from the other. These last are used, for example, to darken the sky without darkening the foreground.