It's not your plant and has nothing to do with watering. It's egg casings left by a critter such as the ones mentioned in the link below.

Here is a link that might be useful: white fuzzy *egg* casings

So the plant is fine and any peppers coming off the plant will be edible?

I'm so new to gardening and I'm glad I found this forum to answer my questions!

Thanks for the help!

Your peppers will be fine to eat as long as you don't use a poison to kill bugs! You can scrape the egg casings off and it'll all be good.

Yep, I've had those before on my peppers. I just wiped them off, and the plants and peppers did just fine. Thanks for posting the link, tomatofreak, that was interesting to read.