Once you've found your app, you'll need to install it on your iPad. This is a two-step process. First, touch the price, which is outlined in red. (Free apps will read "FREE".) Once you've touched the button, the outline will turn green and read "INSTALL APP". Touch it again to start the install process.
When you are on the app's page in the App Store, the download button is located at the top of the screen just above the Details tab. (I've outlined the button on the screen shot above.) You can also install an app directly from the search results by touching the price tag located on the right side of the search results box.
Once you've chosen to install the app, you may be prompted to enter your iTunes account password. This request is designed to protect you in case someone picks up your iPad and attempts to download a bunch of apps without your permission. If you have recently downloaded an app, the iPad will assume it's actually you still using it and won't prompt you for your password. (So if you want to download a bunch of apps at once, the iPad won't ask you for your password each time.)
Want more help downloading apps? This guide will walk you through the process.