When you elope, you get married. The difference between eloping and having a wedding (
AFAIK) is that if you elope, usually the only people there are your two witnesses, and there is no fancy ceremony, and it is often sudden and unexpected (or like you do it because you know your parents won't approve) though a lot of couples do it just because they don't want to deal with a fancy ceremony or a reception.
As for who marries you, if you are married by a justice of the peace, priest, rabbi, or anyone legally allowed to do it, you will be legally married (civil marriage). If you want a religious marriage, then you would not want to be married by a justice of the peace. If you're not really religious at all, these are the same, but some people think that to be married "in the eyes of God" you need to have a religious ceremony.
Anyway, I hope that answers your question. My parents were married by a justice of the peace, and only had a few guests, but it was not an elopement. I dunno, it's just terminology I guess. Typically an elopement implies that you ran off and got married, then came back and told everyone after the fact. Like "Hey mom, guess what I did last weekend? I got married!"