Google Photos is a website where you can upload pictures from your computer and share them with friends and family. There is also basic editing functionality as well as some features that reduce the workload for you organizing your photos collection like face grouping.
Link: photos.google.com
Support site: Google Photos
Google Photos for mobile is an app for use on mobile devices using Android (by Google) or iOS (by Apple). The app offers similar (but not all or always identical) features as Google Photos for desktop.
The app shows:
Photos/videos that are stored on the device (DCIM folder by default and other selected folders)
On Android devices it shows all device folders under "Library"
Photos stored in the cloud for the active Google account
Install from: Google Play on Android, or from Apple app store on iOS.
Google Photos Help: Google Photos
Album Archive is a website that displays an archive of all your Picasa Web Albums and Google Photos, and some more like photos from Hangouts, Blogger etc. More information here
Picasa is a software program that you install on your computer. You can edit and organize your digital photos and videos. Picasa is no longer supported by Google (March 2016). Read more here.
Picasa continues to work on computers, however all online functions were disabled by Google. More info here.
Picasa Web Albums (aka Picasaweb) was a website for photos and videos. Picasa Web Albums was retired in August 2016. Picasa Web Albums were moved by Google to Album Archive and Google Photos. When you delete photos/albums in one of these two places, it is also removed in the other.
Google+ Photos was the "photos" section on the social network called Google Plus (Google+). It basically showed the same as Picasa Web Albums. The photos section was officially discontinued and replaced by Google Photos in 2015.
Google Drive photos
Google Drive once had a Photos sections https://drive.google.com/drive/photos The link now redirects to Google Photos https://photos.google.com/