Single-Servo (AF-S) vs Continuous-Servo (AF-C)

On a visit to Hermanus on the Southern Coast in the Western Cape I have taken a couple of pictures of our daughters on a jungle-gym's swings.  Unfortunately, to my frustration, most were completely out of focus.  Of course I only realized this when going through the pictures when we got home.

I have realized two things that I needed to do:

1. Always have a quick look through the pictures on the viewfinder (especially if you are on location and cannot go back to re-do it).
2. Make sure the Autofocus on my Nikon is set correctly.

Since I haven't had a chance to do the 2nd part of the tutorial on the Nikon D7000 yet, I had a look on Nikon's website and came across this link with more information on the AF-S vs AF-C modes:  http://imaging.nikon.com/history/basics/16/03.htm.

This really clarified to me what the difference between the two settings are and which one is best to use under which circumstances.