What exactly does "Cloud-based eCommerce solution" mean?

This is the term used for the provision of a computer based solution delivered as a service over the Internet. 

For non cloud-based solutions, if you want to set up an eCommerce website, you (or a technical person you hire) would need to set up a server, install software, configure that software to work for your purposes and then maintain and update that software over time. With a cloud-based solution, the system is designed to allow you to quickly create your eCommerce site using a web browser with little technical expertise and have the software behind your site maintained and enhanced automatically without having to worry about technical issues such as backups and software development.