Rainbow cookies or Rainbow cake can refer to any of a number of rainbow-colored confections.
Italian Flag variation
A particular variety of this confection, known primarily by various regional names listed below, is with layers of brightly colored, almond-based sponge cake, apricot and/or raspberryjam, and a chocolate coating. Many bakeries sell them by the pound, and because of the many layers, these cookies tend to be somewhat heavy estimating in about fifteen to a pound. 
This variety of the confection likely originated in Italian communities in the United States during the 1900s, in homage to the community's heritage. Although often called simply rainbow cookies in much of the non-Italian continental United States, some local names for this specific variety are: