In warmer environments, they tend to be called screen doors, and are used when you want to leave the door open without bugs getting in. In areas with bad weather (ice storms and such), storm doors protect the door itself from the elements, especially if it’s an expensive wooden door.

I dunno but I didn't have one for a while and then i did and the temperature difference is noticable. You can also swap out the glass pane for a screen in the summer.

It must be American thing, I have never seen a house with two doors (Europe), apart from front door and back door of course. I am not saying these don't exist but certainly are not common enough for me to see one.