All Weebly sites are hosted on their servers; there is a free basic plan as well as paid plans that provide more advanced features (like E-commerce for setting up an online store).
After using a number of hosting companies and even running my own virtual server to host WordPress sites, there are only two hosting companies I recommend to my WordPress clients:
Economy Solution – SiteGround. As of August 2015, their startup account is about $4.00 per month. What makes them different is, they know WordPress very well and as a consequence, they provide a lot of the security and performance enhancements you’d otherwise have to do yourself. And their support team won’t say “Huh?” when you call them with a WordPress problem. Recommended for sites on the smaller side of the spectrum, with a relatively small number of monthly visitors.
High-Performance Solution – WP Engine. As of August 2015, their plan for a single site is about $29 per month. It’s also scaleable, which means you can upgrade to a higher-end account if you find your site is getting a lot of traffic. Automattic, the company behind WordPress.com, has invested heavily in WP Engine, which should tell you something. An account with them covers everything: hosting, backups, updates, performance enhancements, security, and you can access it all through an easy to use dashboard. This means fewer tweaks and plugins you need to add to your site to address security and performance issues. Recommended for the larger, more complex sites with a lot of content and/or a lot of monthly visitors.
(Note: the links above are affiliate links, which means I will be paid for accounts created with them)