Slow sand filters are a type of sand filter that provide exceptional filtering and require comparatively simple materials and little cost to make. Before you get too excited, this is a DIY project that will take time and will probably be against building and health codes in your area. Slow sand filters are primarily used in developing countries, remote cabins and vacation homes, and occasionally in municipal water treatment plants. There is a specific type of slow sand filter designed by Dr. David Manz called the BioSand Water Filter that is often used by humanitarian organizations to provide clean water in developing countries.
All slow sand filters share one particular feature that could earn them the name biofilter: they all have a layer of life-filled water on top, a miniature swamp full of aerobic bacteria, fungi, and protozoa that filter unhealthy bacteria out of the water (this layer is called a schmutzdecke, German for 'dirt cover'). Because of this layer slow sand filters are always kept wet, with a standing layer of water over the sand. This has traditionally meant that these filters required a steady inflow to remain healthy. Dr. Manz's great innovation was to add a siphon or reverse U-bend, so that water exiting from the bottom of the filter travels up a pipe that arches up just a little higher than the schmutzdecke; the water can only drain through the filter when the inflow raises the water level above the raised U-bend.
Aside from this the filter design is quite simple. Its main body is some sort of water tank, often a concrete box. This box has a PVC pipe leading to an exit valve near the bottom; this pipe is usually covered with some sort of filtering cloth (such as mosquito netting). The tank is filled with a layer of pebbles at the bottom, then a layer coarse gravel, then finer gravel, then a layer of coarse sand, finer sand, and finally a layer of washed sand (It isn't necessary to use all of these layers, of course). More information of sand size and quality is available here.
Once you have this set up, simply start the water flowing through the tank. Let it run for a bit while the water washes out dust and grime from the sand and gravel. Once it runs clear you have a functional filter, but it isn't a true slow sand filter just yet! After it runs for a few days bacteria from the local environment will move into the surface water and start setting up a small ecosystem. This ecosystem will be hostile to most types of bacteria that cause human illness, resulting in a very effective biological filter. It is usually claimed that this sort of filter will get 95-99% of bacteria out of the water, although this is dependent on the quality of the filter. These filters are one of the most effective simple filters available, and will also remove certain minerals (such as iron and arsenic), and reduce odors that may be present in the water.
Because this sort of filtration is not 100% effective it should not be used to make sewage or grey water potable. It is most often used for filtering running water (rivers and streams).
Slow sand filters will slowly 'clog up' as the biological layer on top gets too thick. It must be scraped off and allowed to redevelop, which will interrupt the flow of clean water. This cleaning may be done while the filter is wet, in which case the water present during scraping needs to be diverted from the potable water supply, or after drying out the filter.
Other References:
http://www.biosandfilter.org/biosandfilter/index.php/
http://www.villagehealthproject.org/biosandFilters.html
http://en.wikipedia.org/wiki/Slow_sand_filter