An acid dye is a dye that is typically applied to a textile at low pH. They are mainly used to dye wool, not cotton fabrics. Some acid dyes are also used as food colorants.
Reactive dyes are most commonly used in dyeing of cellulose like cotton or flax, but also wool is dyeable with reactive dyes. Reactive dyes can also be applied on some synthetics like nylon.
Pigment dyeing uses ground pigments, rather than a true "dye." One of the advantages of pigment dyeing is its ability to adhere to a wide range of textiles, not just natural fabrics. This is especially important with synthetics or blends.
Stone washing is a process used to give a newly manufactured cloth a worn-in appearance. Stone-washing also helps to increase the softness and flexibility of otherwise stiff and rigid fabrics such as canvas and denim.