http://tws27.weebly.com/boden-kieseritzky-gambit.html
Although probably unsound, this gambit is better than it looks at first sight, since after 3...Nxe4 4.Nc3 Nxc3 5.dxc3, Black's only way to defend the e5-pawn is to play the weakening 5...f6, whereupon White can generate hacking chances on the kingside with 6.Nh4 threatening Qh5+, and meet 6...g6 by throwing the f-pawn forward (7.f4 Qe7 8.f5 Qg7 9.fxg6 being the typical continuation, leading to considerable complications). Although Black can get the better chances with best play, there is plenty of scope for Black to go wrong (that said, in practice Black scores better than average according to my database).
White can instead defer the attack with 6.0-0, but this allows Black an extra move to organise a defence. Probably best is 6...Nc6, leading to a position often reached via the Two Knights Defence (1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Nc3 Nxe4 5.0-0 Nxc3 6.dxc3 f6). Then after 7.Nh4 g6 8.f4, Tim Harding recommends 8...f5, but I also think John Emms's suggestion 8...Qg7 (in Play the Open Games as Black) is equally good, since White has to attend to the threat of ...Qc5+, and after, say, 9.Kh1 d6 10.f5 Qg7, White cannot play 11.fxg6 safely because after 11...hxg6 Black gets the half-open h-file pointing at White's king. White gets some, but insufficient, compensation for the pawn, but in my opinion Black's position is easier to play than after 6.Nh4, and Black has a plus score in practice.
Two Knights Defence players interested in an analysis from a specifically Two Knights perspective should check out the notes to Game 3 (Fabri-Ashton, Blackpool 2014) which covers 1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.0-0 and 4.Nc3, including the "fork trick" line 4.Nc3 Nxe4 5.Nxe4 d5. It is important to bear this line in mind, since I think White actually has chances of an edge after 6.Bd3 dxe4 7.Bxe4 Bd6 (the standard "book" line) and so Black should look into alternatives at move 7, such as 7...Ne7 intending 8...f5 hitting out at the bishop on e4. Via the Two Knights move order Black can also consider 4.Nc3 Nxe4 5.0-0 Nxc3 6.dxc3 Be7 (instead of 6...f6), returning the pawn, but I hesitate to recommend this because many games with this line end in draws (plus 6...f6 should be better for Black).
From White's point of view, the Boden-Kieseritzky Gambit may appeal to some players- it certainly gives practical chances- but those after a reliable way to sacrifice the e4-pawn would be better off looking into the Urusov Gambit (1.e4 e5 2.Bc4 Nf6 3.d4, which can also be reached via 1.e4 e5 2.d4 exd4 3.Nf3 Nf6 4.Bc4, or 3.Bc4 Nf6 4.Nf3).
For a second opinion, Tim Harding's article on the gambit is well worth checking out.
