Confusion about photo-backups Android 4.4
The Photos app and Google+ app, while deeply connected, are different applications on your phone. The Photos app is designed to replace the old Gallery app by providing more editing and sharing options. Both Photos and Gallery show you pictures stored on your phone and stored on Google's servers under your G+ account. If you're using Photos you can ignore Gallery.

Both the Photos app and Google+ app have Auto-backup settings and both are talking about the same thing: backing up pictures you take to a private album associated with your G+ account on Google's servers. You can turn on Auto-backup in either Photos or Google+.

You can delete or recover pictures that were backed up to G+ easily within the Photos app. From Photos you can also delete pictures that were taken with your phone's camera but not backed up to G+. Pictures that were locally stored on your device or were not backed up to G+ cannot be restored once deleted. Pictures that have been backed up to G+ with have a small cloud icon in the bottom-right corner of the thumbnail.

You can remove pictures just from the device by going to the upper left and clicking on the "Photos".  That should bring up a menu, where you scroll down almost to the bottom to get to "On Device".  Bringing up that list is just the photos on the device (instead of all the photos that G+ knows about), and when you delete there it only affects the device storage, not the backed up version.  See https://support.google.com/plus/answer/3453521?hl=en for a (very) little more detail.