The Ponziani move order is that shown on the board below:
[Fen ""]
1. e4 e5 2. Nf3 Nc6 3. c3
With 3. c3, White aims to:
- Prepare the pawn push d4
- Avoid the Ruy Lopez & Italian game lines, though transpositions can occur
- The down side of this non-developing move is that the queen-side knight is deprived of its best square. White often sacrifices material or structure (by accepting an isolated Queen's pawn) to develop.
In response to this, Black has 3 main moves.
3...Nf6
r1bqkbnr/pppp1ppp/2n5/4p3/4P3/2P2N2/PP1P1PPP/RNBQKB1R b KQkq - 0 3
3...Nf6 4. d4
Now, Black can accept the gambit with 4...Nxe4. Play goes 5. d5 Ne7. 5...Nb8 is also played, but less often than 5...Ne7. White's plan here is as with an gambit - build a development lead, keep the initiative & attack.
Or, he can accept the pawn with 4...exd4. Play goes 5. e5 Ne4 6. Qe2 d(f)5 7. exd(f)5 e.p., opening the position. White often gets an isolated Queen's paw position in these lines, which should be played like a gambit (keep the initiative, avoid trades & attack).
Lastly, Black can play 4...d5, leading to a complex game with lots of central tension. Play goes 5. Bb5 aims to develop & prepare tactics based on the theme of removing the defender.
3...d5
r1bqkbnr/pppp1ppp/2n5/4p3/4P3/2P2N2/PP1P1PPP/RNBQKB1R b KQkq - 0 3
3...d5 4. Qa4
Play can go 4... Qd6 5. Na3 (preparing to attack the Queen) Nf6 6. Nb5 Qe7 7. d3, and White has better development, a space advantage & a solid structure.
Play can also become very complicated with 4... Bd7 exd5 5. Nd4, where Black is trying to develop with tactics against the Queen.
3...f5!?
r1bqkbnr/pppp1ppp/2n5/4p3/4P3/2P2N2/PP1P1PPP/RNBQKB1R b KQkq - 0 3
3...f5 4. exf5 Qf6
This is a gambit with play similar to the Latvian gambit.