In a nutshell -
Dye sublimation is a method of transfering an image - including full colour/ photo images - onto a variety of polyester or polymer-coated substrates.
Unlike other types of prints, dye sublimated image gets DYED INTO the fabric or polymer coating, it's not a print on the surface of the substrate. Dye sublimated print doesn't have any feel to it.
Main advantages - It is suitable for full colour images, as well as short-rins and one-off prints;
Main disadvantages - doesn't work on natural materials and dark substrates.
Dye sublimation is widely used for photo gifts like mugs, bags, mousepads, coasters, stubby holders (and dosens of other products);
A lot of swimwear and sports apparel are printed using dye-sub;
Large format dye-sub printing is a popular way to print flags, banners, advertising displays, etc.
There is a lot of information about this method in dye-sublimation section of the forums.