F-Droid is an archive of free and open-source apps in an official app you can install on your Android. It works similar to how Google Play's own official app work.
The main difference between F-Droid and other app archives is that all its content is open source, no exceptions. In practical terms, this means you won't find any well-known games or 'famous' Android apps in the F-Droid catalogue.
In the F-Droid app you can download any of the included apps, along with these apps' older versions. This app also detects which apps you have installed and gives you the option to update them when possible.
F-Droid is an excellent official app with a fantastic archive of apps for Android. In total you'll find more than 1,500 different apps along with their older versions also available for download. All this in a simple, elegant interface.