What is the difference between folding bead and wire bead tires?
Folding versus wire refers to the material within the bead or 'edge' of each side of the tire. Foldable bead tires use kevlar in most cases (I think) and tires with each type are entirely compatible with the rim size that the tire is designed for (in the absence of a manufacturing defect). Folding bead tires have a few advantages (in my opinion):


Somewhat lighter
Easier to store (can be refolded by flattening the tire and folding it into a bundle)
Bead has more stretch to it, so mounting the tire is usually less difficult


For these advantages, you will probably pay more for a foldable bead tire when a tire model includes both options. Also, wire bead's also appear more often as the OEM tires on lower end bikes.