You're thinking of Nitrous Oxide (N2O). Nitrous Oxide is used during some surgeries (I took it when they pulled my wisdom teeth) and abused by people trying to get high. Nitric Oxide (NO) is apparently a vasodilator (it relaxes blood vessels allowing more blood to flow). It's produced naturally by the body, and you can buy supplements if you really want your pee to have more Nitric Oxide in it (taking most supplements just makes your body get rid of the extra by filtering it through the kidneys).
I don't know anything about the Buteyko method, but it sounds like a fad/scam medical practice with no proven benefits. But you do you, just don't do the tape thing, that sounds dangerous.