In humans, the cuticle is a thin fold of skin growing up over the bottom edge (the lunula) of the finger nails and toe nails. The technical term for this is the eponychium (or, more rarely, the epionychium). It helps keep the matrix from which the nails grow protected and free from infection.
In invertebrates, the cuticle is a multi-layered, non-cellular structure formed from fatty acids with alcohols (but not including glycerol) on the outside of the epidermis. It forms an exoskeleton, but it may be either hard or soft.
In plants, the cuticle is a thin, continuous layer of non-cellular fatty substances, secreted over the epidermis to protect the plant from drying out. Found on leaves, stems, fruits, and other above-ground plant organs.
In inanimate objects, a cuticle refers to a hardened or thickened outer layer, such as a thickened layer on the surface of a liquid, the bloom on an eggshell, or the scaly outer layer of a wool fiber.
'Cuticle' comes from the Latin cuticula, the diminutive of cutis "skin", or "hide". It is pronounced 'kyoo-tik-kul', although my dictionary says that 'kyood-ihk-kul' is also acceptable.