If you're using Twitter on its web interface, i.e., Twitter.com, you'll see your background whether you're looking at your profile or your feed or any other page.
When another user clicks through from one of your tweets on the same interface, they will see your profile page with all of your tweets, avatar, header photo, profile info, and background. We can see most of these elements through mobile and desktop apps as well in one form or another.
You can customize your background in Settings > Design with one of Twitter's preset "themes" or can upload an image of your own, up to 2MB. Or, create your own Twitter background.