As white, I often encounter the Philidor Defense by black:
[FEN ""] 1. e4 e5 2. Nf3 d6
I usually play
d4 on the next move, but is this the best move? What are some lines (advantages or disadvantages) against the Philidor?
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3. d4 is the usual move after
1. e4 e5 2. Nf3 d6. Chess Opening Theory states that it is probably the best move, and also concedes
The best move is probably
3. d4. White threatens a queen exchange with
dxe5 dxe5 Qxd8+ Kxd8and Black can forget about castling. It puts pressure on the center and the Black fortress may collapse at any time.
Another possibility is
3. Bc4leading to a more positional game, playable for both camps.
[FEN ""] 1. e4 e5 2. Nf3 d6 3.d4 exd4 4. Qxd4! Nc6 5. Bb5! Bd7 6. Bxc6 Bxc6 7. Nc3 Nf6 8. Bg5 Be7 9. O-O-O
A good resource to study would be shredderchess.com's opening database.
To complement Daniel's answer: Black often aims for the solid Hanham setup with
Be7 without surrendering the center. The problem is to find a suitable move order from the position
[FEN ""] 1. e4 e5 2. Nf3 d6 3.d4
If black plays
3...Nf6, then white gets pressure and a solid plus after
4.dxe5 Nxe4 5.Qd5!
[FEN ""] 1. e4 e5 2. Nf3 d6 3.d4 Nf6 4.dxe5 Nxe4 5.Qd5!
Black can try
3...Nd7 first, but then the problem is to find a good move after
[FEN ""] 1. e4 e5 2. Nf3 d6 3.d4 Nd7 4.Bc4 (4...c6 5.O-O Be7 6.dxe5 dxe5 7.Ng5! Bxg5 8.Qh5) (4...Be7 5.dxe5! Nxe5 (5...dxe5 6.Qd5) 6.Nxe5 dxe5 7.Qh5) (4...Ngf6 5.Ng5) 4...
4... Ngf6 loses because of
4...Be7 is bad due to the less obvious
5.dxe5! Nxe5 (5...dxe5 6.Qd5) 6.Nxe5 dxe5 7.Qh5
4...c6 is probably best, but loses the bishop pair after
5.O-O Be7 6.dxe5 dxe5 7.Ng5! Bxg5 8.Qh5
The best way to reach the Hanham is probably with the radically different move order
1.e4 d6 2.d4 Nf6 3.Nc3 e5 4.Nf3 Nbd7, although white then has the option of exchanging into a slightly better endgame with
4.dxe5 dxe5 5.Qxd8 Kxd8.
[FEN ""] 1. e4 d6 2.d4 Nf6 3.Nc3 e5 4.Nf3 (4.dxe5 dxe5 5.Qxd8 Kxd8) Nbd7
Against the Philidor you might experiment with a line I developed myself (don't know how good it is), but if they surrender the center (i.e. exd4), then I develop a setup where I fianchetto the dark-squared bishop on b2, play f3 at some early point (with perhaps a later f4). I usually stick the other bishop on d3 then if they chop it with a knight I take back with the c-pawn, getting a better center than Black at the cost of bishop vs. knight. I usually can start up some sort of middlegame kingside pawn majority attack as a result of all this and create mate threats by Qe1-g3 along with the bishop at b2. It seems to work fairly well but I'm sure some IM/GM could refute the whole thing. Not like we regularly play against those guys though. :)