There is definitely a difference between eloping and having a small, non-elaborate wedding. I think Kymberli is describing the latter. Eloping has the connotation that you're getting married on a whim, or in secret, and shows a lack of responsibility. Eloping seems like it's for people who's family or friends don't approve of the marriage, in which case you probably shouldn't even be getting married; if everyone around you thinks it's a problem, you would be a fool to ignore them. It also seems like it's for impatient people who just can't wait to get married. Getting married doesn't mean casting off your friends and family who have been there for you through everything; isn't it worth a little stress to share that special day with all of them who helped you get there?
That being said, I do like Julianna's suggestion of just the man, woman, and preacher on the beach. I think that can be done as long as you're transparent about it. Let everyone know well in advance that you're getting married, but that there will be no "wedding". Letting them know that it will just be the three of you will at least give them the a heads up so they don't think it's some kind of shotgun marriage. Done correctly, I don't think there is anything wrong with it. Done incorrectly and you'll needlessly damage a lot of relationships.