Nice shot at the end.
The opening is called the Smith-Morra Gambit, but this is not how it's typically played (1. e4 c5 2. d4 cxd4 3. c3 dxc3 4. Nxc3). When it's done that way, it's supposed to give a development lead and action for the rooks along the open C and half-open D files.
It's said to be a dubious gambit, and the way your opponent played the first couple of moves was even worse at first glance.
I don't know what to call the opening 1. e4 c5 2. d4 cxd4, but people seem to do it unusually often in response to 1. e4 c5 In any case, the games usually turn out well for me as Black when they do.
Here is an example...and does anyone know it's name and/or merits?