'Arid' is usually used to refer to a climate over a large area, a long-term condition.
'Dessicated' is usually used to refer either to food that has been dried to preserve it or in literature, some work that lacks life and vitality. It is seldom used.
'Dry' is used much more than the other two on a daily basis to describe anything that lacks moisture. It can also apply to a type of humor that is delivered in a calm, seemingly serious manner.