F-Droid is an app store for Android devices. It's an alternative to Google Play that's characterized by the fact that it doesn't require a user account to download applications. Their apps are also open source and come with the source code so that developers can customize them to their liking. While the number of apps available on F-Droid is not as high as in Google Play, the selection is very good and offers a variety of tools.
Open source apps
F-Droid differs from Google Play, where there are free and paid apps, but all with proprietary licenses. In the case of F-Droid, available applications have free licenses like GPL, MIT or Apache. In addition to downloading the application itself, you have the option of getting the source code so you can reprogram it yourself.
Another feature is that you don't need a user account, so F-Droid doesn't store your data downloads or create a profile, as is the case with Google Play.
Thirdly, F-Droid allows you to download multiple versions of an application, so you can avoid incompatible versions or ones that turn out to be inferior to their predecessors, something that happens often but can't be avoided in the case of Google Play applications.
Don't just search, find
The first thing F-Droid shows you is the list of available applications. If you have installed applications that are on F-Droid they'll appear under the "Installed" tab. In any case, if you need to find something in particular, there's an integrated search tool (the magnifier icon).
To customize your search and improve your experience, there are lots of configuration options. Some are disabled because they can be annoying, like applications with advertising or apps that aren't compatible, but which may interest you for other Android devices.
One other downside to F-Droid is that it doesn't have the well-defined categorization that you'd find on the Google Play Store or screenshots of the apps.
What's new in F-Droid
Apps can be directly sent to other devices via Bluetooth and Android Beam (NFC+Bluetooth). A lot of work has gone into repos which now include local hosting options, Tor support, and increased updating support.
An alternative to Google Play
If you're not convinced by Google Play, or you want to expand your list of available apps, F-Droid will offer you a selection of different applications. The apps are free and you can access their source code.