In 1.e4 c5 2.Nf3 e6 3.d4, why does white play and exchange the c5 pawn? If white does not do so, how can I take advantage of this?
As an example: 1.e4 c5 2.Bc4 e6 3.Bb3 Nc6 4.Nf3
what is the best response from black?
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White doesn't always play d4.
Apart from the line you mention, there is also the King's Indian Attack and the Closed Sicilian, where white plays g3 and Bg2, the Grand Prix Attack, where white plays f4 and the Rossolimo, where white plays Bb5 (after Nc6 or d6).
In all those lines d3 is usually played instead of d4.
But 3.d4 is the most common line: It removes the influence of blacks c5-pawn on the centre and installs a knight on a very central square. This leads to a space advantage for white, which can often result in quick attacking victories. Black on the other hand has longterm trumps in his two central pawns and open c-file. This setup leads to strategically and tactically very rich positions, which explains part of the popularity.
Not playing 3.d4 doesn't lead to a worse position for white. It only arguably makes it easier for black to reach an equal position. Even the line you give, which really is unusual, doesn't guaranty black an advantage. He will probably just continue to develop with Nf6, Be7, 0-0 and maybe at one point expand on the queenside or play d5.