The Göring Gambit is a 19th-century style gambit beginning with 1.e4 e5 2.Nf3 Nc6 3.d4 exd4 4.c3!? (or 2.d4 exd4 3.Nf3 Nc6 4.c3).
Instead of capturing on d4 with the f3-knight, White offers the d-pawn as a gambit by challenging Black's d4-pawn with c2-c3. White threatens to play c3xd4 on the next move, creating a classical pawn centre with pawns on d4 and e4, controlling the important central squares c5, d5, e5 and f5. The gambit was first played at high levels by Howard Staunton in the 1840s, but the gambit is named after Carl Theodor Göring, who was the first to introduce it into master play. It is closely related to the Danish Gambit (1.e4 e5 2.d4 exd4 3.c3) but I feel that the inclusion of Nf3 and ...Nc6 improves White's chances in some key variations, particularly those beginning 4...dxc3 5.Bc4.
Although the Göring Gambit has never gained much popularity at the highest levels of play, some players rated in the 2300-2600 Elo bracket have used it frequently at some point, e.g. Jonathan Penrose, Ljubomir Ljubojević, Douglas Bryson, Igor Dolgov and Dimitrios Mastrovasilis. There was a rare GM outing in the gambit in 2011, when Yu Yangyi used it successfully against Rimat Jumabayev. Alexander Alekhine often played 1.e4 e5 2.d4 exd4 3.c3 dxc3 4.Nxc3 in casual games, which often transposed after a subsequent ...Nc6 and Nf3.
If Black accepts the gambit, White will generally put pressure on f7 with Bc4, Qb3 and/or Ng5, and other common motifs include Nc3-d5, and if Black plays ...Ng8-f6 too early then the knight is often hit by e4-e5. White often has to be careful of the ...Na5 fork, which can force the exchange of White's important c4-bishop for the black knight on c6. After 4...dxc3, White can eitherrecapture with 5.Nxc3, which appears fully sound, or offer a second pawn with 5.Bc4, which is riskier but also more dangerous.
If Black declines the gambit with 4...d5, then White often gets an isolated pawn on d4, but aims to compensate by activating the white pieces into the early middlegame.
Black can decline the gambit in various other ways, including 4...Qe7, 4...d3, 4...Nge7 (intending 5...d5) and 4...Nf6, of which the last two are the most important, and probably fall just a little short of providing full equality. However, Black can also be tempted to play passively, with 4...d6 or 4...g6, allowing White a strong central pawn roller with 5.cxd4.
If White uses the move-order 1.e4 e5 2.Nf3 Nc6 3.d4 exd4 4.c3, then White has to be ready for the Petroff Defence (2...Nf6) and the Philidor Defence (2...d6) and the dubious but tricky Elephant (2...d5) and Latvian (2...f5) gambits. After 2...Nc6 3.d4, however, Black has no good way to maintain the strong-point on e5. 3...d6 is well met by either 4.dxe5, simplifying to a better queenless middlegame, or 4.Bb5, transposing into the Steinitz variation of the Ruy Lopez, which is quite passive for Black, or 4.Bc4 which tends to lead into the Hungarian Defence.
If White tries 1.e4 e5 2.d4 exd4 3.Nf3 Nc6 4.c3, then Black has no particularly strong alternative to 2...exd4, although 2...Nc6 (transposing to a line of the Nimzowitsch Defence, 1.e4 Nc6 2.d4 e5) is playable. 2...d6 can lead into a Philidor Defence after 3.Nf3 although White can also consider 3.dxe5 intending 3...dxe5 4.Qxd8+. At move 3, 3...Nf6 transposes into the Old Petroff (here White can consider heading into the Urusov Gambit with 4.Bc4). 3...Bb4+ is a problem for players who use the "double pawn sacrifice" line 4.c3 dxc3 5.Bc4 as a way of avoiding 5.Nxc3 Bb4, since after 3...Bb4+, White's best is probably 4.c3 dxc3 5.Nxc3, and then Black's best is probably 5...Nc6, transposing into the 5.Nxc3 Bb4 line. However, if White is happy to head into that line, then he or she should be happy to see 3...Bb4+.