You need Reflector (it works in PCs and Macs). You don't need to pay for it, cause you can use it for 10 minutes with the free version.
Once you airplay your ipad to your computer, you can also use Reflector to save sessions to video. In my computer it saves it to a .mov, which are usually huge, but I am sure someone could help us find a way to reduce it to something smaller.
I got to record one epic defense I did this am. Enjoy!