Filters:
1) Polarizing filter. Blocks light that's polarized in one direction more than others. This does things like make the sky contrastier, reduces reflections, and plants sometimes look more vivid, if the filter is turned the right way. The sky effects depend on angle from the sun. Be sure to get circular type only for normal DSLRs
2) ND filter. Blocks an even %age of light coming in, so that you can use slow shutter speeds if you need to for creative reasons in bright light, without overexposing the photo.
3) Graduated ND filter. Blocks more light on one side than the other. Used to essentially give the illusion of your camera having greater dynamic range in extremely high contrast sky/ground situations, like sunsets.
4) Color filters block out certain wavelengths of color. If you shoot RAW, you can entirely replicate these effects in software. Although it may be more time consuming to do so than to use a filter, depending.
5) UV filters don't do very much, although in some cases they are legitimately useful if you ACTUALLY need protection in the form of a filter, and a lens hood just won't cut it. Such as salt spray from the ocean, flying sand like at a volleyball game, or in the rain if you're using a weather-rated body and lens (many lenses aren't under warranty as weather sealed completely until they have a filter of some sort added).
4) Various special effects blurring filters. Some give you "glow" around your highlights like flare. Some make it look like there's a misty fog around your highlights, etc.
5) Contrast filters increase contrast by different amounts.
6) Infrared and UV PASS filters. Not the same as a cheap regular UV filter. Instead, these block ALL light except UV or Infrared. They will appear completely opaque black. They allow you to take photos using only light that is invisible to human eyes. However, most digital SLRs have built in IR and UV blocking filters, so if you use these filters, you're blocking IR for instance and also blocking everything except IR... which means almost no light gets into the camera. Usually the lens filter wins the fight, though, and with several minute exposures on a tripod, you can still get an IR image or UV image. But do research on this before buying these expensive filters.
7) Variable ND filters. Same as an ND filter, but you can rotate them, and change the amount of light they block. Cheap ones can make a distracting "X" shape on your image.