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Originally named because the initial moves mirror those of the King's Gambit, only played on the queenside rather than the kingside, the Queen's Gambit is not really a gambit at all, since White can regain the 'sacrificed' pawn immediately if desired. If Black accepts the pawn, it is almost invariably with the intention of allowing White to recapture it within a few moves - see Queen's Gambit Accepted. Historically, Black has preferred to decline the offered pawn, maintaining a strong central position instead.
2.c4 e6 (There are other ways to decline the offered pawn, for example 2...c6, the Slav defense, but they do not fall into the category of the Queen's Gambit Declined opening)
These opening moves already demonstrate the intentions of both players. White is going to try to put pressure on the center of the board and expand on the queenside, and Black is going to develop on the kingside and maintain a strongpoint in the center with the d5 pawn. If at any time Black captures on c4, White will regain the pawn quickly and then organise his pieces to force the pawn push e2-e4, giving him an advantage in the center. A typical continuation:
White's long-term plan in this kind of position is generally the minority attack, in which the queenside pawns are pushed forward in an attempt to break up Black's solid pawn structure and create weaknesses which can be attacked. In the meantime, Black is going to build up forces on the kingside and attack White's king, hoping to create enough threats to keep the game balanced.
The Queen's Gambit Declined was immensely popular during the Classical period of chess history, when players tended to focus above all on maintaining a strong center, following the classical theories of development and pawn structure, and not doing anything silly. It is still popular as one of Black's most solid replies to a White queen's pawn opening, but many players now favour more active, dynamic defenses such as the King's Indian Defense or the Benko Gambit if they are interested in trying to win with Black.