I'm wondering how I could record the screen of my iPhone without jailbreaking? Are there any Apps that will do this? I know how to take a screen shot but I don't know how to record video.
I don't believe there's a way to do it without jailbreaking. Because of the security restrictions on what apps can access, and particularly the limited functionality when running in the background, I'm pretty sure it's impossible without jailbreaking (and maybe even if you do jailbreak).
If you're looking to make a screen-cast or something similar, the technique I've heard is popular (at least with Adam Lisagor, who does a lot of iOS app videos) is using the iPhone simulator included with Xcode.
Since iOS 8 and OS X Yosemite, you can connect the iPhone to your Mac using the Lightning connector, and it will be available as a video camera. Then you can record video of what happens on the screen using QuickTime Player.
You can use an OS X app like Reflector or AirServer to make your Mac simulate an AirPlay display. Then you select your Mac as an external AirPlay display on your iPhone. AirServer and Reflector provide features to record the output.
OS X 10.10 Yosemite has an iOS screen recording feature built into QuickTime.