The Internet is the best tool in human history for finding information. Of course, depending on the information in question, you might not want it found. Whether it's an embarrassing picture of you, an angry status update or stupid video you posted, you might want it gone.
Unfortunately, it's often hard to remove information from the Internet. Even if the original is taken down, copies might still be scattered around, and those copies are just a Google search away.
That's not Google's fault by the way; it's just pointing to information that it finds. However, because Google is the go-to place to find things, many people think it controls what appears on the Internet.
The European Union seems to agree. That's why it ruled that people have the "right to be forgotten." In other words, if a European asks Google to stop displaying certain links in a search result, Google has to do it.
While some U.S. companies are starting to adopt European privacy laws, it's unlikely the "right to be forgotten" will be available to U.S. citizens anytime soon. Yet, it is still possible to remove your past from Google's search results. You just have to go about it a different way.