Hi,

For Odoo there is different versions available, the main classification is the Odoo Enterprise edition and the Odoo Community edition. The Enterprise edition is the paid one and the community edition is free one. The Odoo enterpise is the extra set modules installed over the odoo community edition.

To know more difference between the odoo enterprise and the community see this: Odoo Enterprise vs Odoo Community

For Odoo community you can download it from the Github and install it locally or you can host it in your own server and use it. There is no limitation or restriction for it. You can use customize the system by adding new modules or editing the code itself(not recommended one).

For the Odoo enterprise edition, we can again classify it. Enterprise edition can be installed in on-premise server or you can get cloud based system like odoo.sh or odoo online. In the odoo online the customization is only possible via odoo studio, no custom modules can be used with it. To overcome this drawback of the odoo online, odoo recently introduced the odoo.sh, in which the custom code can be used.

For better understanding of the difference between the odoo online and odoo.sh, see this link:  Odoo.sh Vs Odoo Online

Thanks