No connectivity is required, it's TOTP on a 30-second interval. The tl;dr: is that you have a shared secret (so if this ever gets leaked to an attacker, yes, you're vulnerable) which is used in conjunction with current time (give or take a few seconds) to generate a code you can use to confirm authentication.
This may sound silly, but keep in mind that TOTP requires that both ends agree on the current time. I learned this the hard way when my authenticator stopped working consistently.
Apparently I had disabled my device's (the one with the authenticator app) "automatically set time from NTP" feature. Over time this resulted in my device's clock drifting X seconds away from the providers' clock(s), which in turn resulted in my occasionally using codes that were already X seconds expired.
The counter based OTP is actually more secure, but Google doesn't go for them with end-users, because they can go out of sync (eg if your kid is idly flicking through a lot of them on your phone) and then have to be reset.
I have an original iPad. It doesn't get connected to the net. So, no way to get the app on there, and it probably wouldn't work for iOS ~5.1.1 anyway. ;)
That being said, it hadn't clicked that a non mobile (eg laptop/desktop) version of it could exist.
The wikipedia page for it says it's strictly mobile only[1], as does the Google install info page[2].
So (for me) it's a real PITA when places require a mobile phone number and there's no way to skip it. Obviously, can't use those services.
Does anyone know if Google Authenticator would run on a wifi iPad? As a potential workaround for the "no mobile network" situation.