How to record your screen on your iPhone without jailbreak?

Are there any apps for this?

Press the power button and the home button together

You mean print screen/screen capture? You press the home button and the lock button at the same time.
EDIT: To record? Um, I don't think that's available at the moment. You can probably use another phone's video camera, but for the time being, I don't think there's a way to record like it's been screen captured…
Sorry.