It all depends on how you have your continuous focus priority set. If you want to use continuous shutter, then to get focusing, you have to set the menu to focus priority. By default, continuous focus priority is set for release.
AF-C is for continuous shooting. If you have this set to focus priority release, it will focus on only the first shot and the rest will not focus.
AF-S is for single exposures. If the focus priority is to focus, then it will run autofocus on every single shot.
I'm not sure if you meant exactly what you wrote here, but your statements are very misleading, if read literally.
Focus Priority is not required in order to have the camera perform focusing. You still have AF working when Release priority is selected - it is just that the user is able to take a photo when the AF system doesn't think the focus is within its "window of perfection."
This can be important in action photography, where the timing of the shot needs to be very precise, to catch a position or moment where a slight mis-focus does not matter as much as a slight timing miss would. Using focus priority in action photography can be annoying, to say the least, because the camera is quite finicky about having focus perfect before it allows the shot to be taken.
AF modes, in order of preference (best first):
AF-C only means continuous focus, not continuous shooting. It is useful for both bursts and for single frames, and gives you full-time subject tracking. If you have your camera set up for back-button focusing, there is no need to use any other AF mode.
AF-S means the AF will lock, i.e., stop operating after the camera thinks good focus is achieved. It is most useful for focus-and-recompose, but could also be used for burst shooting if you do not want the camera to refocus during the burst. Personally, I never use AF-S unless I am in a focus-and-recompose situation, but even then, the alternative of using back-button focusing may be better. The camera may perform subject tracking in this mode, if it thinks the subject is moving, but my experience is that this cannot be counted on.
AF-A allows the camera to make its own decision about whether AF-S or AF-C is most appropriate. This is only for users who don't understand AF-S and AF-C. It can be rather unpredictable, and the fact that it isn't offered on the pro cameras should tell you something. I call this mode AF-Avoid.
