I only want to back up photos and not videos.
In the official Google Photos app, you can't disable backup of videos per se, but you can exclude folders having videos from being backed up. That can be done like this:
In the "Device Storage" view (accessible from the swipe-in sidebar), look for the "cloud" icon to the right of the folder name.
If the Cloud icon is blue with an up arrow on it, backup of that folder is enabled. Tap on the icon to disable backup for that folder.
The Cloud icon should now become white with a slash through it. Congratulations! Backup of a folder has been disabled successfully.
It might happen that your videos are intermixed with your photos, especially if they have been shot on your camera. Move the videos to a separate folder using a file manager app, usually "Videos", and disable backup for that folder from the "Device Storage" view.
This will disable Google Auto-Backup for the selected folders only, leaving the rest unaffected.
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Image Credits @ mobilesiri.com