There are several ways to host an Odoo database

Odoo online

This is Odoo’s main hosting option. Your Odoo database is on a shared instance with a URL that will be something like <yourcompany>.odoo.com

There is no additional charge for hosting, but the main limitation with Odoo Online is that you cannot install third-party apps or do your own development.

• However, some limited customization can be done, either using Odoo Studio or by manually creating Models / fields, adding or extending views, creating menu options, etc.
• Automated Actions and Server Actions are also available.

With Odoo Online, you can have one app free but if you have more apps you will pay both for the users and the apps, as explained here.

Odoo.sh

This is a more sophisticated hosting option which does allow third-party apps to be installed. It offers integration with GitHub for development.

You cannot have a free app, and of course you have to pay for the hosting (based on the number of Worker(s), the Storage and how many Staging environments you need).

App pricing is the same as for Odoo Online.

Self-hosted

Of course you can host Odoo on your server, or on a cloud-based server. Here again there is no free app and pricing is the same as Odoo Online. So it is probably fair to say that Odoo Online hosting is free, and the explanation is that Odoo believes support costs are lower when they have full control over the instance and do not allow third-party apps or development.

Odoo Community

Another option is to use Odoo Community. No Odoo hosting options are available, but you can have as many apps and users as you like for free. There are many options for hosting, including some companies that have Odoo-specific offerings.