Chess Opening Theory/1. e4/1...e5/2. f4/2...exf4/3. Nf3/3...g5/4. h4/4...g4

White's knight is attacked. Although there are some King's Gambit positions where White can ignore this attack and carry on developing, this isn't one of them. 5.Nd4 is just going to give away a tempo to 5...Bg7, and 5.Nh2 undefends h4 with dire consequences.
 * 5. Ne5 is the Kieseritzky Gambit, which can be considered the main line of the whole King's Gambit. White targets f7 and g4, in the hope of recovering her lost pawn and leaving Black with weak pawns on f4, f7 and h7. But while that knight is going on a tour of the board, Black is catching up on development with moves like ...d6 and ...Nf6, and even if Black does end up with doubled isolated f-pawns, the one on f4 is going nowhere and remains a thorn in White's side.
 * 5. Ng5 is the Allgaier Gambit. Black can now trap the knight with 5...h6, forcing it to sacrifice itself on f7 to displace the Black king. How well this typically works can be illustrated by the fact that the Kieseritsky Gambit is statistically one of the most terrible openings in the White repertoire, but it's still about ten times as popular as the Allgaier Gambit.

Theory table
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1.e4 e5 2.f4 exf4 3.Nf3 g5 4. h4 g4