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There are two ways to access the App Store: directly from your iPhone or iPad (using the App Store app that comes preinstalled on your device), or using iTunes on your primary computer. The App Store app is used exclusively for finding, purchasing (if applicable), downloading, and installing apps directly on your device from the App Store; iTunes on your primary computer is used to access the App Store, but, among many other things, can also be used to access the iTunes Store and Apple’s iBookstore to find, purchase (if applicable), download, and install many other types of content.
As you’re about to discover, some apps available from the App Store are free. To download them, you still go through the same process as you do for purchasing an app; however, instead of tapping on the Price icon, followed by the Purchase icon to confirm your purchase, you’ll tap on the Free icon associated with the app, followed by an Install icon. You will not be charged for downloading a free app. You will, however, still need to supply your Apple ID password to confirm the transaction, download, and then automatically install the app.