Installation and subscription

All the major apps like Photoshop, Illustrator, Lightroom and Lightroom Classic, etc., download and run on your desktop computer.

All of these desktop apps (except Lightroom, which maintains your photos in the cloud) will save the files locally on the same computer, or you also have the option in many programs to save your files/projects as cloud documents (which are available from anywhere).

Creative Cloud also offers many mobile apps for iOS and Android which typically integrate with the desktop apps and the cloud.

Lastly, the notion that these "are online apps" is a common myth, although now Adobe is starting to develop versions of Photoshop and Illustrator that will run in a web browser:

https://helpx.adobe.com/photoshop/using/photoshop-web-faq.html

However, this is entirely optional to use, and just another way to access the apps available in your subscription.

Normally, you can use your desktop apps offline and Internet access is only needed occasionally to recheck your subscription status.

Here is more information about how often you need to connect to the internet:

https://helpx.adobe.com/creative-cloud/kb/internet-connection-creative-cloud-apps.html

As mentioned by Axel Matt, what apps would you be using?

It depend on which application do you use. For example, if you work with PhotoShop and Ligthtroom classic you can work complete locally. The apps are install on you hard disk and your files (photos, catalog, etc.) can be save on your local hard drive. You can synchronize it with the cloud but you must not.

If you work with Lightroom CC the photos are saved in the cloud.