I don't mean to bring up a few day old topic, but I feel I should clarify @mfitz804. My family owns a business that does silk-screen printing, heat pressing, embroidering and (unrelated) engraving.
Screen printing or silk-screening is the process of transferring ink onto the shirt through a mesh screen. What you described above is actually heat pressing, in which a pre-cut or pre-printed material, usually paper or vinyl, is applied to the shirt using pressure and heat. The commercial version of this is basically a better version of iron-on materials. In my experience I have found that screen printed designs typically last longer than heat pressed ones, though they can still fade over many washes, while the heat pressed designs to crack or peel eventually. In addition, the screen printed ink doesn't have the same thick or plastic-y feel and is usually more flexible and comfortable.
@Nathan's description of sublimation is spot on, as the dyes are actually injected into the fabric of the shirt. Your in/on analysis is quite correct, @mfitz804. I believe sublimated designs will last longer than both screen printing and heat pressing.
EDIT: Sorry, I tried to quote the original post and tag the appropriate parties, but it doesn't look like any of that worked.