F-Droid is an installable catalogue of FOSS (Free and Open Source Software) applications for the Android platform. The client makes it easy to browse, install, and keep track of updates on your device.
The F-Droid app? Its not distributed on GPlat officialy. Could be a trojan.
On an unrelated note I would say I trust F-Droid more than GPlay too. Don't have GPlay or Aurora atm
Download from the link instead. Once you download the apk file you can install it straight from your phone. You probably need to give permission for it to be installed.