Understanding focus modes (AF-S, AF-C, etc.)
The focus mode controls whether autofocus locks after finding focus or keeps updating while you track a subject. It does not decide where in the frame the camera should focus. That is the job of the focus area or focus point.
Still subjects
Single autofocus, commonly called AF-S on Nikon, Sony and others or One Shot on Canon, is designed for subjects that remain at roughly the same distance. The camera focuses, then holds that distance while the focusing control remains engaged.
Moving subjects
Continuous autofocus, commonly AF-C or AI Servo, keeps recalculating focus while you hold the focusing control. It is normally the better starting point for sport, children, pets and other subjects moving towards or away from you.
Names vary
Manufacturers use different names, and some cameras add automatic modes that switch behaviour for you. Use the manual for your exact model rather than assuming another brand uses the same labels.