How to tell which application has pushed a notification
Just long-press the notification. That should either directly lead to the corresponding app in Settings › Apps – or at least reveal a button to go there:


The ⓘ button brings you to the details, where you also can configure notifications for that app (Android 5.1 here)

AFAIR that feature was added somewhere around Kitkat (Android 4.4), but not before JellyBean (4.2), so it should definitely be available with Android 5 (Lollipop).