What is the technology used to prompt for passwords on an public wifi network
As far as I know, it is not called "Push notifcation" as indicated in earlier answer.

The correct name for this is a "Captive portal".

The technology is simply that the user's requests are redirected to a specific web page until the user has authenticated (or whatever requirements the specific provider has).

The redirection can technically be done via HTTP (i.e. some kind of HTTP proxy), via ICMP redirects or via sending back fake DNS replies.

Apple has added detection of these captive portals to Mac OS X. After connecting to a WiFi system, Mac OS X will try to access an address at Apple - if it succeeds it assumes the user has general access to the internet.

If not, it will assume a captive portal is in place, and popup a window for the user to authenticate to the network.