When working with leather or suede, the best adhesives to use are polyurethane glue, contact cement or cyanocrylate-based glue. These glues are available from a variety of manufacturers, with several types of applicator choices.
Polyurethane glues, such as Gorilla Glue or Titebond, yield good water resistance by the tight bonds they create when the foamy texture expands into the gaps in the leather or suede in the presence of moisture. However, shoemakers often prefer contact or rubber cements, such as G-S Hypo Fabric Cement or Barge All-Purpose Cement, both for their fast evaporation rates and the flexible rubber seal left behind when they dry. Cyanocrylate-based glues, commonly known as super glues, are particularly long-lasting in leather work, since they don't dry out as quickly over time, and are also noted for non-discoloring properties.
If is also possible to use one- and two-part epoxy glues, such as E-6000, on leather or suede, but these adhesives are less flexible, and as such are best reserved for adhering metal findings or gemstones.
As for applicators, squeeze tubes and brush-on applicators are best for covering large surface areas, while users can handle small tears or detail work with drop bottle or push point applicators.