Mod is a shortening of modern—and perhaps modernist, a term used to describe a player or aficionado of modern jazz. In 1960s Britain, mod identified a young person who dressed stylishly in tailored suits or miniskirts and rode around on a motor scooter listening to the blues and to soul music—rather than the heavier rock music emerging at the time. By association, the name came to be used as an adjective for fashionable people and for things stylish and trendy (in other words, modern).
Modish is also an adjective meaning "fashionable" and "stylish," but it is a much older one (entering English in the mid-1600s) and unrelated, being based on the French word mode, which denotes a way of living, thinking, or dressing.