BlackBerry 10.2.1 introduced the ability to install .APK files, which are the app files used in Android, directly to your phone. This throws the doors open to install a wide variety of Android apps on your BlackBerry 10 device, be it a Z10, Q10, Q5, Z30, or anything else. This used to be possible through a long, convoluted process, but it's much easier now. Basically, if you can't find an app in BlackBerry World, odds are there are other sources for your favorite app.
You can get an Android APK install file on your BlackBerry by visiting the source in your BlackBerry browser (which is the easiest option), e-mailing yourself the file, or moving the file directly from your computer to your phone by plugging them together with a USB cable.
If you downloaded the APK from your browser, the Downloads window should have already popped up asking you where you want to save it and what the filename should be; just tap the file when it's done. If the download window isn't there, it's easy enough to get back up. Tap the overflow menu button in the bottom-right (three little dots), and tap Downloads.
If the file was moved to your phone in any other way, you can find it easily by tapping the magnifying glass icon from the home screen at the bottom in the middle and typing ".apk" in the search box at the top. Anything you've moved over should show up under the Documents and Files result. Just tap the one you want to install.
Your Android app should start installing after a short "Processing" indicator. Tap the Install button in the top right, and tap Accept on the pop-up window which informs you of the device permissions that are required for the app to run.
The first time you install an .APK, you'll be prompted to turn off a safety setting which otherwise blocks installation of these files. The previous installation screen should return with a handy "Open" button replacing the "Install" button when it's done. At that point, the app will now show up alongside all of your other apps. Just swipe on the home screen from right to left a few times to find the icon and put it in whatever folder or home screen you like. Hold your finger on the icon for a moment to drag it around. Another thing you may want to do is hide the bar at the bottom of the screen. Just swipe down from the top when the app is open, and tap Hide bar.
Now, remember that there are no guarantees that these apps will function as advertised. BlackBerry does not run Android, it can just run Android apps. The Android app player in BlackBerry 10 is good, and should handle most of what you throw at it. Test things out and let us know what's working and what isn't in the comments! Having trouble installing something? You can ask for help in the comments too!