@Lauren,
Places your applying at aren't just going to take your word on it. Theey will ALL do some kind of background check. But in your case, I'd say that works out great for you (well better than just filling in the form). If you were caught selling drugs 15 years ago, and have had a successful business, and have been completely out of all related trouble since then, you should not be worried much.
First, next to the check box where it asks if you have been convicted of a crime/felony/whatever, it also says please explain. Thats a perfect place to put a one line response explaining the circumstance.
Also, because of the background check, they will see that its 15 years old, and that you've had your own business since then. This will go a LONG way. I'd say you got nothing to be worried about, unless of course your applying for a job at a school, daycare, etc..
good luck