The solid material with the lowest known density is called aerogel. It was original invented by chemist S.S. Kistler in 1931 as part of a bet. The article reporting his discovery is available online. The idea was to somehow remove all the liquid from a jelly without it shrinking. As detailed in his paper, Kistler was able to achieve this by first replacing the liquid in the jelly with another with a low critical temperature, such as alcohol. The gel was then heated under pressure so that the liquid turned into a gas without evaporating. The gas was then allowed to escape without disturbing the (aero)gel structure. The result was a material with an incredibly small density and some remarkable physical properties.
One possible use of the material, reported by Scientific American in Feb. 2009, is as a super-sponge. Because the material has so much surface area and internal space available, it can absorb several times its weight in liquid. It has been considered for use in the cleaning up of oil spills, as well as for treating wastewater. (Perhaps you have an aquarium that uses an activated carbon filter. Like aerogels, activated carbon has a large surface area and an affinity for organic pollutants.)
The material also makes an excellent insulator. In order for heat to be transferred through a material by conduction, the molecules of the material must be able to interact with each other to pass the energy along. (Your thermos has a layer of vacuum insulation to prevent precisely this.) Because aerogels are such tenuous materials it takes a long time for interactions to transfer heat through them. According to one report, some homes and clothing have already been insulated with aerogels, with excellent results. (I suppose it isn’t quite practicable yet to vacuum insulate your home!)
But, the most famous use of the aerogel thus far has probably been in space missions. The NASA Stardust mission captured cometary dust, using aerogel to gradually slow the dust particles without damaging them. Aerogel was also used on the Mars Pathfinder rover.
Finally, it should be noted that aerogel is not one single material, but a class of materials that have all been similarly prepared and feature the same extremely low density. Aerogels based on silica are most common, being blue in color and quite cool to look at. If you’d like to see some aerogel in person check with your local science center. I know there is some on display at the St. Louis Science Center, as well as at the Smithsonian Air & Space Museum in Washington, DC.