Update on 2016/05/01: Version 2.1.6 is available. This fixes the download problem. Please update to this version, then log out current device and log in again.
Update on 2016/04/28: I’m aware of the downloading problem. I’m looking into it now. Sorry for the inconvenience.
Update on 2015/12/19: Google decided to take download APK Downloader on their Chrome Webstore (after featuring it on the store for a year), so from now on you’ll have to install/update it manually. Sorry about that.
The latest version of Official APK Downloader is here. Now supports Firefox
APK Downloader is a Google Chrome extension / Firefox addon that allows you to download Android APK files from Google Play Store (Android Market) to your PC