I don't know anything about the 1st Generation AppleTV, but the AppleTV2 is the only one that I know can run a web browser-- as I have a browser on mine. I know for a fact that you cannot put a browser on an ATV3, as there is not, and will not be a jailbreak for it for quite some time.
But really, why would you want one on an AppleTV? A browser is quite useless on these devices. Your browsing is limited, and your typing/input is a pain in the @SS. No real advantage with it. I have a browser on my ATV2, but never use it, as it is more pain than gain.