I installed a GIF app from Play Store, and they want permission to do this. I don't even understand what drawing over other apps means.
Is it the floating hearts in a GIF in a private message in Messenger? I'm confused.
The technical name of this permission is
Allows an application to open windows using the type TYPE_SYSTEM_ALERT, shown on top of all other applications. This permission allows an app to show a "popup" window above all other apps, even if the app is not in the foreground. A malicious developer/advertiser could use it to show very obnoxious advertising. Almost no apps should require this permission unless they are part of the Android operating system. An example of a system alert would be the alert you are shown when your phone or tablet is out of battery and is about to shut down.
As GiantTree already pointed out, this is also used by "screen overlay apps" such as screen filters. A malicious app could e.g. use it to trick you to do things you'd otherwise wouldn't do, for example overlaying buttons with different text (or replacing the question text with something different) so you press "yes" on a screen you wanted to press "no" (so-called "Tap-Jacking") – which is why (at least with Android 6) people often get the warning "screen overlay detected". See e.g.:
GiantTree is right. If the "floating hearts or animated movements" are supposed to appear on top of other apps, then this permission is needed. If that's not what it does or if that's not what you want (i.e., if it's only supposed to be for editing GIFs inside the app), then you should remove/not install the app — such overlays can be used to intercept input and perform unwanted operations on your phone or steal your data.
A classic example of a valid use for this permission is, again as GiantTree mentions, screen filtering. Such apps can darken the screen or red-shift it for use at night. Another example is Facebook Messenger, which can put chat icons on the screen to make notifications more visible and provide quick access to current conversations.
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