Well, in its simplest form, a paper piercer is a handheld tool that looks like a large needle with a handle. It works a lot like a hole punch, but on a much smaller scale. The purpose of a paper piercer is, as its name indicates, to pierce paper, but instead of creating an eraser-sized hole (like a hole punch), a paper piercer makes a pin-sized hole.
A lot of scrappers will use it to make a hole in their cardstock when attaching brads because brads by themselves aren't sharp enough and have a tendency to rip your precious scrapbook paper. A paper piercer will make a nice pin-size hole in your paper for the brad to fit in. You can also use it to make starter holes for thread work, if you like to embellish your scrapbook layouts with fancy thread or ribbon.
You can find paper piercers at about an arts and crafts or hobby store and even on Amazon. They're inexpensive and a must, in my opinion, in any scrappers tool kit.