You can download the same app as long as its compatible with the iPad and you are signed in with same user account. When you signed in and found the app on your iPad, you should have seen the button marked with "install" rather than "buy". Also, you should be able to see your past purchases in the AppStore. If that's not the case, you are likely not syncing the devices to the same Apple account.
There are also iPad or HD versions of many apps that are separate purchases. Many apps are "universal" and run on many devices, but not all.