It's a bit of online software designed to create a mobile friendly site. Quote from the Weebly blog:
I've just optimised one of my sites, and it's not bad at all, especially given that I spent less than 5 minutes on it. I could, if I wanted to, spend a lot longer and do a lot of editing. The only downside so far as I can see is that your domain name gets a 'dudamobile' stuffed into the middle of it, but that seems like a fair trade off for a free and easy mobile site. The other downside is that it's supposedly only free for a year, but I shan't worry about that right now.The service essentially comes from DudaMobile, which is all about auto-generating mobile sites from your standard site. dudamobile costs you $9 a month for the premium plan. But if you sign up through Google’s gomo initiative, it is free for one year. Gomo does not need any payment details (no credit card info needed). So, you do not need to worry about getting charged accidentally after an year.