Italian game purpose/advantage


I'll try to answer by contrasting it with the Spanish development (Bb5). I play 1.e4 as white and 1.e4 e5 as black... but... in trying to sum up some of the richest openings chess in a few sentences hopefully I don't make a fool of myself haha
In the Spanish, as white, you're playing d4 and (often) getting positions where black can't play d5 (in most cases if black can successfully play d5 he's equalized)... so you have some space / center advantage that you leverage to play in the center / kingside.
In the Italian you're postponing the decision on d4, sometimes just leaving your pawn on d3 the whole time. It's a bit slower, but you're more flexible and can react to black's setup.
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For example in the Italian, with the help of the pawn on d3, you may try to absorb their play in the center or queenside while attacking their king on the kingside. The Italian bishop on c4 is on the great attacking diagonal a2-g8, pointed right at the king. You can also choose to expand in the center (c3-d4) or on the queenside (c3-b4).
Often black chooses to play Be6 to trade off white's nice bishop. Similarly Be3 by white when black has Bc5. This move isn't required, but it defuses any quick <25 move mating attacks.
The Italian is an opening that can lead to some big attacks, but again, I'd summarize it by saying it's a bit slower, more solid, and more flexible.

Not to be too much of a dork, but some positions I imagine like a hoplite soldier... a shield in one hand and a spear in the other.
In the Italian the d3 pawn buttresses the center (shield) while the bishop on c4 aims at the kingside (spear).
That's probably a nice quick explanation for a new player
#1
"just wondering what’s the purpose of the moves." The purpose of 1 e4 is to open the diagonal for Bf1 and to prevent 1...Nf6 in view of 2 e5. The purpose of 2 Nf3 is to develop the knight, attack pawn e5 and prepare O-O. The purpose of 3 Bc4 is to develop the bishop attacking f7 and to prepare O-O. There are many ways to continue, but the present fashion is to continue d3, O-O, Nbd2, c3, Re1, Nf1, Ng3 and gradually prepare d4 and/or an attack on the black king.
It is very similar to the Ruy Lopez 3 Bb5 a6 4 Ba4 ...b5 Bb3. The difference is that in the Ruy Lopez white has the option to trade Bxc6 and also that in the Ruy Lopez the black pawns have advanced a6, b5. The latter may be an advantage e.g. to develop Bb7, but also a disadvantage as white may break with a4.
The old masters like Anderssen preferred the Italian. Then the Ruy Lopez was preferred by Lasker, Capablanca, Fischer, Kasparov. Then the Italian became fashionable again by Kramnik, Carlsen, Caruana. Now the Ruy Lopez is again more fashionable.
At the Yekaterinburg Candidates Italian was played 4 times: 2 draws, 0 white wins, 2 black wins. Ruy Lopez was played 10 times: 6 draws, 4 white wins, 0 black wins.