You might use Mastodon and the rest of the Fediverse through your server's website or through an app, but there is also a third option that many people prefer: web apps.
Web apps are special kinds of websites that can also be installed like apps. They appear as app icons, keep you signed in, work separately from your browser and provide notifications just like any app does.
Web apps are extremely easy to install, there's a complete beginner's guide at:
https://fedi.tips/how-to-install-web-apps-for-mastodon-and-the-fediverse
A lot of app store apps are actually just web apps packaged to be downloadable from app stores!
But web apps don't give Apple or Google any money or control, so they tend to get downplayed. It's also harder to access a user's data from a web app than an app store app, so companies who want to spy on their users do it through app store apps.