If someone wanted to design tattoos and show them for sale, how could they keep people from stealing them?

You can't. It can known as a certain artists work first, but if an artist uses a design given to him that you draw it'll be HIS work because he inked the person. So you may have came up with it but basically unless you're a tattoo artist it doesn't matter because people won't know you drew it anyway. However, if you are an artist or are going to become one you can copyright it when you're an artist and then it'll be known as your work.

You can't, really. There is copyright law, but that gets complicated.