King's Gambit Modern Defence: 1.e4 e5 2.f4 exf4 3.Nf3 d5
With 3...d5, Black aims to return the gambit pawn and concentrate on development. Unlike many other such declined variations, the Modern Defence tends to lead to a relatively dynamic equality, rather than sterility, and so is a good option for those who would rather avoid the wilder lines of the King's Gambit Accepted but still retain reasonable winning chances. I don't think it is a strong reason for White to give up the King's Gambit though, because there is still a fair amount of scope for White to head for unbalanced play with attacking chances. Note that Black can head for the Modern Defence with 2...d5 3.exd5 exf4, which avoids the line 2...exf4 3.Bc4 d5 4.Bxd5.
White should continue with 4.exd5, since 4.e5?! should be met by the change of plan with 4...g5! intending 5.d4 g4, and White ends up with an inferior version of 3...g5 lines. Then if Black continues with 4...Qxd5 then White can get some advantage with 5.Nc3 harassing the black queen, and 5.d4 is also quite good. Therefore Black should meet 4.exd5 with 4...Nf6, intending to recapture on d5 with the knight.
4.exd5 Nf6 5.Be2 and 5.d4
5.d4 has little independent value and is inferior to most of White's alternatives. White prepares to recapture the pawn on f4, but then with 5...Nxd5 Black grabs the d5-pawn and defends the f4-pawn. Then White's best is probably 6.Bc4, transposing to the line 5.Bc4 Nxd5 6.d4, but in that line 6.0-0 is superior. 5.Be2 is more interesting, and has been tried by Hikaru Nakamura. The most critical response appears to be 5...Nxd5 6.c4 Ne7, intending ...Ng6 defending f4, and then White has to demonstrate sufficient piece play to compensate for the sacrificed pawn. I am not sure if it is enough, while John Shaw in his book on the King's Gambit does not believe in White's compensation.
5.c4, defending the d5-pawn, appears to lead to dynamically equal play and is thus a reasonable option for White. Black normally responds with 5...c6 (although 5...Bd6 is also fully playable) 6.d4 cxd5 7.c5 Nc6, and now 8.Bxf4 promises equal chances for both sides. This line is not generally very popular, but it has been tried by Magnus Carlsen.
White generally intends to bring the black queen out into the open with 5...Nxd5 6.Nxd5 Qxd5. The main problem for White is 7.d4 (7.c4?! Qa5) 7...Be7, which leaves Black with at least equality and maybe a little more, the key idea being 8.c4 Qe4+. If 6.Bc4 then Black has 6...Nxc3 7.dxc3 Qe7+, intending 8.Be2 g5, and Black generally gets the upper hand. Some of the sidelines are quite attractive for White, but objectively 5.Nc3 is one of White's weaker options.
Causing some disruption in the black queenside before Black can get around to taking on d5. White gets a slight advantage against 5...Nbd7 and 5...Bd7, and against 5...c6 6.dxc6 bxc6, so Black must continue with 5...c6 6.dxc6 Nxc6, which keeps the chances level. Probably best is 7.d4 Bd6 8.0-0 0-0 9.c4 Bg4, with equal chances. The rival pawn majorities give both sides reasonable scope to play for a win, and unusually for the King's Gambit, White will generally attack on the queenside and Black will attack on the kingside.
Traditionally White's most popular move, which aims to give White the majority of the kingside attacking chances. White's idea is generally 5...Nxd5 6.0-0, since after 6.Bxd5 Qxd5 White will struggle to demonstrate sufficient piece activity to compensate for conceding the bishop-pair. White gets good play against 6...Be7, with either 7.d4, or 7.Bxd5 as tried by Magnus Carlsen, which is better than the bishop-for-knight exchange at move 6. However, 6...Be6 is more of a problem, and then 7.Bxd5?! Qxd5 is now good for Black, since if 8.d4 Qf5. White therefore normally retreats the bishop with 7.Bb3, whereupon 7...c5 (Nakamura-Adams, London Classic 2011) is problematic for White. Contrary to John Shaw, I think that White can probably get dynamically equal play with 8.d3 or 8.c4, but both 8.d4 and Nakamura's 8.Kh1 are unappealing for White.