"Piano" means "quiet" in Italian, but your position can turn into a meatball if you don't understand this popular opening. Most every chess player will play this opening at some point, often with White and with Black.
This is another "e4/e5 opening." The main difference between the Giuoco Piano and the Ruy Lopez is that White's bishop develops to c4, which pressures the weak spot, f7. In the Ruy Lopez, it goes to b5, targeting the knight and weakening e5 as a result.
In order for you to play the Giuoco Piano as White, you have to develop your king's knight first. If instead you play 1. e4 e5 2. Bc4 Nf6, then 3. Nc3 allows the crazy variation beginning with 3...Nxe4. In you play 3. d3 instead, Black can play for a quick ...c6 and ...d5. You could play the Morphy Gambit with 3. Nf3 Nxe4 4. Nc3, but if Black plays perfectly you won't get enough for your pawn.
So, if you are taking the time to study the Giuoco, let's play 1. e4 e5 2. Nf3 Nc6 and then 3. Bc4. Now 3...Bc5 is the Giuoco Piano (3...Nf6 is a major alternative -- the Two Knights Defense, and we are covering that in another chapter).
After 3...Bc5, we have a big decision to make. Do we play "really quietly" with the Giuoco Pianissimo -- 4. d3 -- to simply continue normal development? Or do we go for gold and play 4. c3, trying to play for d4 and rule the center? My preference is on the second choice -- why not see what Black is made of!?
The point of 4. c3 is to play d4 very quickly, and we need a third defender on that square. We also sometimes can bring our queen out to b3, which would make a battery leading to the all-important f7-square. So what's the drawback? Well, not only does it fail to develop a piece, it also takes away our b1-knight's best square, which makes our e4-pawn a little weaker also.
Black has some tricky replies to our plan of d4. If he does nothing active, like 4...d6, then we grab the center with 5. d4. If, however, he tries a move like 4...Qe7, you can't play 5. d4 immediately, because he will trade pawns and play ...Qxe4+. The remedy is simple: castle with 5. 0-0 and pretty much no matter what he does, play 6. d4. Now after d2-d4 he can never win your e-pawn with his queen since you'll always have Re1, pinning the queen and king, or at least you'll have dxc5.
Also, if Black tries 4...Qf6 (with an x-ray attack on d4), you can actually play 5. d4 anyway! That will shock your opponent! After 5...exd4 6. e5 is the stong idea. Any queen move is answered by 7. cxd4 with coplete control of the center, while 6...Nxe5 is met by the pin 7. Qe2 and White has too many threats (Qxe5+ and the fork cxd4).
Black's best reply is probably 4...Nf6, aiming at that pawn that we can't guard anymore. Now if you guard with your queen, on c2 or e2, you won't have enough defenders when you play pawn d2-d4, and that was the whole point of 4. c3!
Don't forget your plans!