I know those two words don't really go together--eloping isn't so much a rules game. But they might if you are a fully-grown adult who no longer makes decisions for parental effect (OK, those qualities can coexist). I guess what I'm saying is that the reasons people "elope" are diverse and guidance regarding how not to offend might be a help.
I use the word elope to mean getting hitched without a proper wedding, but I know it's full of other innuendo: young lovers separated by an unyielding parental situation for instance, a wedding that has to happen because they can't live without each other or because there's a bun in the oven. So exciting!
I have a feeling decisions to elope often have less romantic motivations. Drunken short-sightedness, among the lusty young, perhaps. Or not having the money and sedative supply to pull off a three-ring circus style wedding. For those unaccustomed to bucking tradition, it might help to know that large weddings are a fairly modern phenomenon, with weddings outside the home becoming commonplace in the last century, and receptions in any form even later. Far more traditional than a 200-plus person ordeal is the presence of immediate family and close friends.
For couples on a tight budget, putting together a traditional wedding/reception might require narrowing the guest list to an uncomfortable degree. Excluding everyone instead of hurting some feelings is a way some couples solve that problem. Another benefit to doing things quietly: the control factor. Big weddings typically involve a many financially and emotionally invested parties with a variety of interests. This is a phenomenon my fiancee refers to as "too many strings on fingers" (does he mean too many things to do, like string around the finger reminders? No, he waves his fingers in front of my face like a puppeteer--this is an allusion to marionettes). The impossibility of pleasing everyone can become a huge, unwelcome distraction. I handle such stress with dissociation. Eloping can put the focus back on the main event and eliminate the necessity of a zombie-state to get through your own wedding.
One more reason: most people will understand. It's common knowledge that weddings can be an enormous expense and in tough times, an extravagance. Eloping is understandable to most, and probably to everyone who has ever planned a wedding. There are many ways you can include friends and family in your celebration, even if only you, your fiance, and two bums you pulled of the street to bear witness are present during your exchange of vows. You may want to inform certain parties beforehand--parents, siblings, or anyone else whose blessing is especially important. You can still send out engagement or marriage announcements, participate in any of the traditional festivities surrounding a wedding you choose, have a simple party instead of a formal reception or delay the celebration until it's more financially feasible.