Can I record my FaceTime session?

Not directly on your iPhone/iPad, but you can record it on your computer (Mac or PC).

http://www.imcapture.com/IMCapture-for-FaceTime/

Note: I've read that people have had trouble using this on Lion, but I can't speak to that.

Good luck

While video is playing, open quicktime (free), choose "record movie."

All it records is the audio and me -- I want the video of the person I'm talking to.

Select the "New Record Screen".

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