Paper piecing is sort of like fabric applique only with colored cardstock instead of fabric. You cut the pieces of cardstock and piece them together to make a figure to be the focal point of your page.
I start with a simple coloring book picture of a character I like. Pooh would be a good one to start with. Choose something to start with that has a limited number of colors and without a ton of detail. Avoid people until you've practiced. Faces and hands can be difficult, but they're not impossible. Print out your picture. This is your pattern.
Then cut it apart into the different colors. For example, Pooh's red shirt, Pooh's gold face, white eyes, black pupils, etc. Sometimes I use pens to do eyes if they are small. Decide which parts are going to overlap the others. The underneath parts are going to need tabs added to the puzzle pieces to allow for overlap.
Now, trace the puzzle pieces onto colored cardstock. I draw the folds in Pooh's shirt, etc now so if I lose the pattern piece it's no big deal. Overlap and glue into place as you go. Sometimes I use a little yarn or other embellishment to give it more depth.
This sounds a lot more complicated than it really is. These make the cutest Disney pages. And there is a lot of cute Disney clipart and coloring books out there. Some designs can be made really fast!
I can show you a few I've made (I'd have to look for them and I can't guarantee the picture would be that good since they're in albums) or you can do a search on ebay to see them. People are always selling them. Some are much better than others.