Normal paper -- including notebook paper, newspaper, construction paper -- is all made out of cellulose, which comes from trees. The trees are chemically broken down into their individual wood fibers, and the cellulose fibers are chosen and formed into very thin sheets to create paper.
Paper money, on the other hand, is made from paper made of rags. Cotton or linen fabric is beaten to create cotton or linen fibers. You have probably heard of "rag paper" or "fine linen writing paper." This is where it comes from.
It turns out that rag fibers bond together much more firmly than fibers in regular paper. Rag fibers are basically unaffected by water, whereas cellulose fibers absorb water and come apart when they get wet. So paper money comes through the washer just fine, while cellulose paper comes unglued.
Here are some interesting links: