Visit Sheldon Brown's website for a good explanation on tyres.
Folding tyres have a kevlar reinforced bead as opposed to a wire reinforced bead. Unless the tyre has been designed for tubeless set-up like some MTB tyres, it requires an inner tube. Folding tyres are generally the upper end of the road bike or mtb tyre market.
My experience with them is that yes they are good, for low rolling resistance and higher tyre pressures, but if you want a hard wearing compound or a puncture resistant road tyre, then wire reinforced clinchers are adequate. For example, Continental make a tyre called the Gatorskin which is hard wearing, kevlar reinforced with a wire bead and is therefore non folding. You can twist them into a figure of eight for a short while, like home from the shop but not recommended to leave them in this position.
Singles are glued onto the rim and are also foldable. Because they are glued to the rim, they do not need a bead like a clincher tyre, and can therefore also be folded. They are difficult to repair due to the tube being stitched into the tyre. If you get a puncture, you have to rip the tyre away from the rim. The domain of serious racers or track cyclists. They run much higher tyre pressures and as a result of this and the carcass threads per inch (TPI) they have very low rolling resistance. Sheldon Brown will explain all.