I am a new member, so I'm not sure if I am allowed to post an outside URL, but one site that can be helpful in getting started in familysearch.org. When you go there you may want to start with your grandparents or someone before that to have a better chance to find someone.
Be very careful with this site, especially with the "Ancestral File" entries. There are many, many incorrect entries. You are told to verify the information, but many ignore it, thinking that simply because it is posted, it is right. But the entires are not verifed at all. As an extreme example, try following the British royality line back and back-as far as you can. One line actually goes back to Odin-the Norse God!
Rich
ps-Please-this is not a putdown of the site-you can get a lot of GOOD info here, but be sure to verify everything from another source.
You can even try [(surname) origin] in a Google search (no parentheses or brackets, of course). Depending on the name, you may be pleasantly surprised with what you find.
Here is another good one-GenCircles. Like most sites, you can search by surname. But on this one, you can also do it another way. If you share your information by uploading a gedcom file to the site, (once everything get processed) it will do something they call "smartmatch." It will search and then show you individuals who appear to be the same person from every other file on the site.