To "tap" in this context means to cut threads into a hole. For this topic, we can think of three basic things a screw can do - drill its own hole, tap its own threads, or just force its way into the material tearing out a hole or threads in the process.
Using this image from that wikipedia article:
The top screw is self-drilling. The sharp, split point acts as a drill bit to create a properly sized hole. At the top of that drill point, the threads taper up, and this is the self-tapping portion. All self-drilling screws are also considered to be self-tapping, and the common vernacular is to just call them "self tappers".
The next three screws in the picture have a blunt tip, so they are not capable of drilling. They require a pre-drilled hole, but they have a notch or flute at the tip, and this is what the screw can use to cut its own threads or "self-tap". These are pure "self tapping" screws, but you don't see them for sale at the big home stores. They are used more in industry to assemble goods on a factory level. You can buy them - they're just not as common.
The Wikipedia entry has it wrong
This picture from the Wiki page is just regular sheet metal screws and shouldn't be on a page describing self tapping screws:
These do not have a drill tip, and they do not have a flute to cut threads. Sure they are a little sharp, but not sharp enough to make a hole in sheet metal. They need a properly sized hole. If driven into a hole that is too small, they will tear the hole wider (if they are stronger than the sheet metal) and create a weaker connection than a self-tapper that properly cuts the right threads.