By playing c4, white "sacrifices" his piece. However, if accepted, white can gobble control of the center and hope to win, even a pawn down, with that control. However, it is not a true gambit, as the pawn can always be taken (black defending it would result in poor development)
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After World War II, it was less frequently seen, as many Black players moved away from symmetrical openings, tending to use the Indian Defences to combat queen pawn openings.
The Queen's Gambit is still frequently played and it remains an important part of many grandmasters' opening repertoires.
With 2.c4, White threatens to exchange a wing pawn (the c-pawn) for a center pawn (Black's d-pawn) and dominate the center with e2-e4. This is not a true gambit, as Black cannot hold the pawn, e.g., the following line: 1.d4 d5 2.c4 dxc4 3.e3 b5? (Black tries to guard his pawn but should pursue development with 3...e5!) 4.a4 c6? 5.axb5 cxb5?? 6.Qf3! winning a piece.
The Queen's Gambit is divided into two major categories based on Black's response: The Queen's Gambit Accepted (QGA) and the Queen's Gambit Declined (QGD). In the QGA, Black plays 2...dxc4, temporarily giving up the center to obtain freer development. In the QGD, Black usually plays to hold d5. Frequently Black will be cramped, but Black aims to exchange pieces and use the pawn breaks at c5 and e5 to free his game.
Technically, any Black response other than 2...dxc4 (or another line with an early ...dxc4 that transposes into the QGA) is a Queen's Gambit Declined, but the Slav, Chigorin Defense, and Albin Counter Gambit are generally treated separately. In fact there are so many QGD lines after 2...e6, that many of them are distinctive enough to warrant separate treatment. The Orthodox Defense and the Tarrasch Defense are two important examples. See Queen's Gambit Declined for more.
There are many other possible responses:
The Slav Defense is a solid response, although many variations are very tactical. If Black plays both ...c6 and ...e6 (in either order), the opening takes characteristics of both the Slav and the Orthodox Defense and is classified as a Semi-Slav Defense.
The Chigorin Defense takes the game away from the normal positional channels of the QGD, and has been favoured by Alexander Morozevich at top level; it appears to be playable for Black.
The Albin Countergambit is a sharp attempt for Black to gain the initiative. It is not common in top-level chess, but can be a dangerous weapon in club play.
The Symmetrical Defense is very rarely played. Although it has not been definitely refuted, play seems to favor White.