Italian Game. (why c3?)

It prepares for d4 so that if and when d4 is taken, you can take back with a pawn, and you again have 2 pawns in the center.  Was this like a trick question or something?

Mainly, it prepares d4 and plays against the knight on c6 by controlling b4 and d4. White argues, that having the knight not blocking the c-pawn gives him quicker controle over the centre with c3+d4

It also helps to open up the second long, white diagonal down which B & Q can attack the Black King's "fortress" after castling. One problem - Nc3 is no longer possible... but d4 allows Nd2.

I'm concentrating on Ruy Lopez at the moment, and c3 is also a popular move there. 

The databases indicate that an *immediate* move 5.d4 is appropriate -  5 ... Nxe4 looks nasty but I think it's an empty threat.

Maybe someone more learned than me can tell you exactly the moves for white to make after ... Nxe4. I've had some success playing 5.d4 and winging it after 5...Nxe4, though only tried it against < 1200 players.

I play this with black so I am very good prepared against this.

Yes, but immediate logic tells you that e4 needs to be protected, the N can't do it because c3 has  taken its square, therefore you have to move d3.

So in trying for e4 it appears you are forcing yourself to move d3. This is highly illogical... Of course there is a deeper logic that means you *can* move immediately to d4, but I doubt anyone without supernatural combinatorial vision could see *immediately* why.

The Ruy Lopez is usually thought of as superior to the Italian game for equally subtle reasons. That is, in the main line, you end up with exactly the same situation, except in the Ruy Lopez black has moved his g and h pawns to drive back the B to b3, weakening the Q side, and leaving the B on a better square (b3, protected by a pawn...) 

the idea of playing d3 and later d4 is to be flexible and wait for a good moment to push d4. if you push c3 and d4 right away black is just as well prepared for opening the position as white and white got nothing

Yes, sorry, I did mean e4 - I have corrected in the original post. 

According to my database 14 000 players moved 4.c3, to 5000 moving 4.d3, with ultimate victory for white 6% better with c3. OK not a great difference. c3 and a few other moves are perfectly OK.

But my experience is that around the 1200 level, in 15:10, you see people falling into traps after you play c3 - like taking e4 with N. Beginner's reasoning would be that e4 is not protected, so (hey) why not... then you attack the N with castled rook, Q, etc. It's got me a few quick wins in 15:10, and it's not a bad line "in expert play", in fact it's the main line!