@ penguin, it's Gelato - but the quality one finds in Italy is far better - higher cream content, less over-run, etc.
Then there are granitas (without any dairy products) = simply frozen fruit-flavored sugar syrup with a somewhat fine ice crystal structure but not as fine as a sorbet.
On top of that there are slushes / slushies
, which are more of an icy-cold (frozen) beverage that is stirred (liquified?) to a consistency that one can drink through a straw but still quite (icy) cold.
there are a few other frozen summe-time treats, like ice popsicles, etc.
@28, the terms ice cone or shaved ice may vary across the US, like soda, pop, or soda pop. It seems to me shaved ice is served more frequently in regions with larger Asian populations. Mention of shaved ice brings memories of some super-yummy toppings, like grass jelly, condensed milk, and sweetened red beans, to name a few, that are sprinkled on top of the ice. But it seems unlikely a snow cone vendor would offer those types of toppings. Never know.....
Regardless, same concept - flavored finely shaved ice.