Once you've found your app, you'll need to install it on your iPad. This is a two-step process. First, touch the gray button with the price on it. (Free apps will read "FREE".) Once you've touched the gray button, it will turn green and read "INSTALL APP". Touch it again to start the install process. (Watch Video)
When you are on the app's page in the App Store, the download button is located on the left side of the screen below the large icon. (I've circled the button on the screen shot above.) You can also install an app directly from the search results by touching the button located on the right side of the little search results box.
Once you've chosen to install the app, you may be prompted to enter your iTunes account password. Apple will ask for this password if you have not recently downloaded an app.
