Red pepper can refer to any variety of pepper that is red, whereas cayenne pepper is a specific type of red chile pepper. Crushed red pepper, commonly sold as a spice, is generally a blend of dried red bell peppers, cayenne, and ancho peppers.
Cayenne pepper originated in the Cayenne region of French Guiana in South America. It has a distinct, slightly smoky flavor and a moderate spiciness compared to other chile peppers.
Bell peppers are relatively mild, while poblano chiles (called anchos when dried) are spicier. The seeds in crushed red pepper are often thought to give it extra spice, but it is the veins in the flesh of the pepper that have the most kick.