'Cape Canaveral' is actually a few different things. One is a geographic feature; it is an offshore island to the east of Merritt Island, off the eastern coast of Florida. Another is a city; the city of Cape Canaveral is on the cape itself, south of the canal cut to serve Port Canaveral, north of Cocoa Beach. The third is probably the most famous - Cape Canaveral Air Force Station. This, specifically, is the United States Government space launch facility built on the island of Cape Canaveral. Both government and commercial space launches have used and continue to use this facility; it contains a relatively large number of launch sites for various types of vehicles. The Mercury and Gemini manned space programs launched from Cape Canaveral.
The Apollo program was given an entirely new facility, just across the Banana River to the west, on Merritt Island. American manned spaceflight moved there with Apollo and continued to operate exclusively from there during the Space Transportation System years (despite plans, the military Shuttle Launch Complex at Vandenberg AFB was never built). This facility was named the Kennedy Space Center, in memory of President John F. Kennedy who had set the United States on the path to the Moon with Apollo. Kennedy Space Center (KSC) is separate from Cape Canaveral Air Force Station and is not in fact located on Cape Canaveral, although the two facilities obviously operate quite closely and share many resources.
KSC has only ever had two launch sites, Launch Complex 39A and Launch Complex 39B. Both were used for Shuttle operations, including those memorable occasions when two shuttles occupied the pads. Cape Canaveral AFS has four currently active Launch Complexes as of early 2016, with literally dozens of older inactive sites.