The modem is a Speed Touch Home.
wow, I used one of those 8 years ago I did not know they still being used. If I remember correctly it did not have integrated pppoe or pptp client so it requires you to run a pppoe or pptp client on your mac. Therefore when connecting it to an apple airport basestation you have to configure the pppoe client in the base station.
so open your airport utility select your base station , choose manual setup
the click the 'internet' icon in the toolbar of the airport utility window and from the connect using drop down choose PPPoE, then configure it with your isp login details. The click the update button.