Simply go into Settings and tap General>Accessibility>Guided Access to turn it on and select a 4-digit passcode (that you’ll have to remember to get in and out of the locked screen).
The next time you open the app (or start the video) you want to run for them, triple-click the home button and choose which restrictions you want in place. I usually circle the top row and bottom row of the screen so that all of the touch-screen options (like “share” and “pause,” etc.) are off-limits. This keeps him from accidentally pausing or exiting an episode of Caillou every five seconds. Or firing off random tweets. Or deleting personal photos.
To stop the session (even if it’s just to adjust the volume), you triple-click the home button again, enter your passcode, and select end. You can use the triple-click/guided access trick any time, once you’ve set-up the option in Settings.