Re: Google gallery vs new Google Photos
Photos that are saved locally are on Internal Storage, usually in the /storage/sdcard0/DCIM/Camera directory. The system's Media Scanner periodically scans the entire storage area for media files (photos, videos, music) and adds those files to the gallery or music databases so that they show up in whatever Gallery app or Music app that you use. So the files aren't actually stored in those apps--the apps just know where to look to access the files.
The stock Gallery and the Google+ Photos apps usually can access the same photos, whether they're saved locally or in your Google+ Photos cloud. However, on some phones, the stock Gallery can only show locally saved photos.
In the Photos app, you can specify to view only locally saved photos by swiping in from the left and tapping Device Folders.