I never knew what this really was until last Friday. Turns out it's ice cream with soda poured over it (like, duh!). The Italians have affogato (vanilla ice cream with espresso coffee). The Americans have ice cream soda - mine was vanilla ice cream with root beer. And very good it was too. In St Francis Fountain, San Francisco with actual booths too.
Correction: I am informed by a genuine American that this isn't an ice cream soda. So I'll just have to go back and have one of those as well.