Summarize:

This is a blue app that has a white "A" with a white circle around it. It's most likely on your iPhone's or iPod's Home Screen. It's on the bottom of the screen. You can also tap an app on the "Featured" page, which is the default location to which the App Store opens, or you can tap Categories or Top Charts left of Search to look for apps based on different search criteria. It's at the top of the screen. Doing so will take you to a page listing any apps that match your search query. You can also tap the app's name below the search bar to search for it. If the app isn't free, you'll tap its price (e.g., $1.99) instead. If it's an app you already own but isn't on your device currently, you'll tap the download button (the icon looks like a cloud with a down-arrow) instead. This green button will appear in the same location as did GET. You may not always see this option for free apps, but you will always need to verify your account details before purchasing apps.  You can also scan your fingerprint if your iPhone supports Touch ID. If you aren't signed into your Apple ID, you'll also need to enter your Apple ID email address. Once it finishes downloading, the app will be on a page to the right of your iPhone's or iPod's Home Screen. You can tap OPEN in the space where GET was to open the app directly.
Open the App Store. Tap the Search tab. Tap the search bar. Type in the name of an app, then tap Search. Tap GET to the right of your app. Tap INSTALL. Type in your Apple ID password. Wait for the app to download.