One of the bad things about the Internet is that information is hard to remove. That's also a good thing sometimes, but I'll go into that another time.
In other words, that embarrassing picture, angry status update or stupid video you posted might never go away. Even if the original is taken down, copies might still be scattered around the Internet. Worse, 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. Unfortunately, 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 recently 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.
Due to our laws in the U.S., that isn't likely to happen here. However, it is still possible remove your past from Google's search results. You just have to go about it a different way.