The iPad's built-in gyroscope automatically detects when you turn the tablet from a horizontal landscape orientation to a vertical portrait one, and adjusts the screen display accordingly. If you are not seeing this behavior, you may have the orientation lock switched on, which fixes the display in one position. Another possibility is that you're running an application that doesn't support landscape or portrait mode, and so rotating the device won't make any difference.
Double-tap the iPad's Home button to bring up a row of recently used apps. Swipe left until you reach the media player controls. If the icon on the far left side shows a circular arrow symbol with a padlock inside, tap it once to remove the padlock. If there is no padlock, the orientation lock should be off. If the icon shows a speaker symbol, the orientation lock on your iPad is controlled by the side switch, in which case flick the side switch upwards or to the left to disable the lock. Check that no padlock symbol is shown by the battery charge level on the notification bar.
Turn the iPad from a vertical position to a horizontal one by rotating it in your hands. Your hands should be at the shorter ends of the tablet -- by the Sleep/Wake button and the Home button -- rather than at the longer sides. Most iPad apps, including the official Apple software programs and iOS itself, should rotate to match the new orientation, provided that the orientation lock has been disabled.
Reset your iPad if the device isn't responding to the new orientation. Press and hold the Sleep/Wake button for at least 10 seconds. Ignore the red "slide to power off" message and keep the button depressed until the Apple logo appears, which indicates that the device is restarting. This clears the iPad's memory and resets any frozen apps without affecting your data or files.
If you're still having issues with the iPad's orientation after performing these steps, contact Apple Support (link in Resources). While the majority of iPad apps are designed to respond to changes in the tablet's orientation, some apps -- such as those that have only been optimized for the iPhone and iPod Touch -- are fixed in one particular mode. This approach is sometimes used in games where the dimensions of the display must remain fixed at all times. Check the documentation supplied with the app or visit the app's official website for more details.