This is defined as an ecommerce site that is hosted on your own web hosting account. No, I don't mean on your home computer (you guys were supposed to leave!). I mean, you buy web hosting from a good web host like HostGator, Bluehost, etc.
The, you install ecommerce software on that hosting account. Typically this would be open-source ecommerce software like Magento (my favorite) or OS Commerce. A good web host has tight integration with these open source solutions and makes it pretty easy to set them up.
Pros - Typically, lower monthly costs…you're just paying for hosting and merchant fees. Integration with additional website functionality like a blog is more seamless, customization and design options are pretty much wide open. Only limited by your skills and/or budget.
Cons - This is a steeper learning curve. You have to set up everything yourself or pay someone to do it. Despite a good host making initial setup a breeze, further customization can be challenging. Support for the ecommerce platform may be limited to forums or wikis.