The Ponziani Countergambit is Black's most violent way of treating White's Ponziani Opening. White plays 3.c3 in order to support the later d4 push. As Black you have a few very playable options to answer 3.c3.
The three Main Lines in this position are 3...Nf6, 3...d5 and 3...d6. These continuations account for the vast amount games that reach this position. The Ponziani Countergambit, 3...f5, is the most important sideline. It can have a powerful psychological effect.
White will usually play 4.d4 here. This is answered by 4...fxe5. 4.d3 is another less popular way to decline, played to hold the e4-pawn. Here 4...Nf6 is the usual reply. White can also accept the gambit with 4.exf5 although this is also a sideline. You will push the e-pawn to kick the Knight if you get this move.