What does grounding an electrical plug actually do (and why do we ignore it)?

Take a look at Apple laptop plugs: The two different ends plug into the exact same adapter, but one has three prongs and one has two. And anyone can buy those two-to-three converters that seemingly unground the third prong.

What’s going on here? How does it work, and why is it both important enough to be kept as a standard, yet no one ever thinks twice about ignoring it?