With Heads-up, any device can get the heads-up notifications introduced in the Android 5.0 “Lollipop”, no root required. But according to Addictivetips, this app is much more than just a backport. And according to the users, this app is way better than the original!

Heads-up displays the latest notification as a floating popup, complete with full text and quick actions (4.1+).
Works on everything running Android 3.1 and newer.
You're in control: Heads-up has five pages of customization, so you'll get notified your way.
Pocket detection – when you get a message, Heads-up turns the screen on if you take your phone out of the pocket! (Kind of like AcDisplay. Note: this causes a tiny increase in battery usage if you get a ton of notifications)
Privacy on the lock screen (and in general - Heads-up doesn't even have internet permission, so it can't send your private data anywhere)
Oh, and it's known to work on some smartwatches.

Heads-up is not a lockscreen app, so don't expect it behave like the Lollipop lockscreen.

Heads-up notifications is available in over 20 languages, thanks to an awesome team of translators. As this app isn't supported anymore, new translations will not make it into Heads-up.

If no popups appear, please make sure that Heads-up shows up in the list of running services. If it doesn't, make sure you have enabled the service, and then reboot the phone.

If your device is running Android 4.2 or earlier, and you have other accessibility services running (such as Pushbullet), they might block Heads-up from receiving your notifications. Please try to disable them if Heads-up doesn't work.

Huawei, XIAOMI and MIUI users: please allow Heads-up to display popups under Settings > Apps > Downloaded > Heads-up. It might also be necessary to reset the app data/settings.

This app uses the Device Administrator permission. It is used to work around a bug where the screen doesn't turn off on certain devices, and you should not grant the device admin permission unless you are affected by this bug. If you find that the uninstall button is disabled, this is because you have granted this permission. To revoke this permission so you can uninstall the app, go to System Settings > Security > Device administrators > Heads-up, and tap Disable.

Other: Facebook Messenger, AcDisplay and PushBullet notification mirroring might conflict with Heads-up.

Heads-up is open-source, licensed under GPL!
That means you can make your own version of Heads-up, as long as the resulting app also is open-source.

I don't have time to work on this app anymore, and don't expect answers to questions about the app. Sorry :(
From version Heads-up Notifications 1.12.1:
Hi everyone!
As you might have noticed, I don't really have time to work on Heads-up anymore. Heads-up will keep working, but I won't add any new features. Maybe someday in the future :)
I'd like to say thank you for using Heads-up, leaving over 30,000 awesome reviews here on Play, sending me thousands of emails with feedback, and translating it into 30 languages. And of course a huge thank you to those who donated!
I'd never be able to pull this off without you!

All versions
February 23, 2016