Dye sublimation technology is used for a wide variety of products, from apparel to phone cases to mugs. A special printer deposits a special dye on to a film. The film is placed on the surface of the product and heated; the heat causes the dye to diffuse into the product and dye the material.
Because the product material is dyed (rather than depositing ink on its surface), this produces incredibly vibrant prints that are not easily damaged by washing. However, this only works with material that readily accepts dye. Cotton is not commonly used; DYESUB apparel is almost always 100% polyester or other synthetic fabric.
Another advantage of dye sublimation is that it allows printing anywhere on the product. The printer prints on a film that can then be applied anywhere on the garment; the garment itself does not have to fit in the printer or on a rectangular platen. This allows for "all over prints". And of course, on curved surfaces like mugs.
Advantages
Vibrant colors that feel a part of the fabric rather than "on top of" the fabric
Print anywhere on the product, including "all over"
Disadvantages
Requires synthetic blanks that accept dye
Expensive for fabric, which is usually cut apart, dyed, then sown back together