In other words, all your apps are listed in the Appstore under "purchased." If there is an app on that list that you do not have on your iPad, you can press the cloud symbol in the app cell and it will install the app onto your iPad. If you bought another app with another appleid, you have to logoff your current appleid and log back on using the other appleid.
As an example, I have three iPads. One is my first iPad and over time I have bought many apps. You can see in the picture below, I have 181 apps (YIKES). BUT not all of these apps are on this iPad, you can tell which app is on this iPad, by the word 'installed'. I have a second iPad that I use for my small business. If you look at the picture below, you will see business type apps. There is a cloud in the app cell. If I press this cloud, I will get to the install screen for this particular app. I can then install the app on this iPad. BTW, this is how people in the same family share apps across two iPads.
In summary, your backup to the iCloud is for primarily data files: music, photos, e-books, documents-->original to you. Things like apps and iOS are not back upped because you can find it elsewhere in the Apple world.