Many 1.e4 players are frustrated by the Petroff Defence, 1.e4 e5 2.Nf3 Nf6, whereupon if White grabs the pawn on e5, 3.Nxe5, then 3...d6 follows, encouraging 4.Nf3 Nxe4. However, in the Cochrane Gambit, White instead speculatively sacrifices a knight on f7: 4.Nxf7.
I was inspired to take a look at this line after examining David Bronstein's dubious piece sacrifice in the Two Knights Defence (1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Ng5 d5 5.exd5 Na5 6.d3 h6 7.Nf3 e4 8.dxe4 Nxc4). The Cochrane Gambit, despite looking crazy at first glance, is rather sounder because White gets a similarly strong centre, and also leaves the black king misplaced. Black often spends time getting the king into a relatively safe position which allows White to build up a space advantage. Rather than aim for a quick attack against the black king, White generally seeks a slower build-up, aiming to crush Black by pushing the central pawns forward, although many lines also give White long-term attacking prospects against the black king.
After 4...Kxf7 White generally continues with 5.d4, since 5.Bc4+ is met by 5...d5. If Black plays 5...Nxe4?! then White wins the knight on e4 with 6.Qh5+ g6 7.Qd5+, or 6...Ke7 7.Qe2. My examinations of the line have suggested that the line 5.d4 g6 6.Nc3 (if 6.Bc4+ Kg7 7.Nc3 Be7) 6...c5 may be the hardest line to crack, where Black aims to put the king on g7 and play to undermine White's d4-pawn, encouraging a pawn swap on c5 and a queen swap on the d-file.
In most of the other lines White appears to be able to get sufficient compensation for the sacrificed knight without much difficulty. For example, 5...d5 6.e5 Ne4 7.Bd3 leaves Black's knight unstable, 5...Be7 6.Nc3 c6 7.Bd3 Re8 8.0-0 is promising for White, and 5...Qe8 6.Nc3 c5 (suggested by John Watson) is fine for White after 7.Be3!? (a Rybka suggestion) and 7.Bc4+ Be6 8.d5.
5...c5 is tricky, since 6.Bc4+ is met by 6...d5 7.exd5 b5, and 6.Nc3 g6 transposes to 5.d4 g6 6.Nc3 c5, so White's best is probably 6.dxc5 Nc6 7.Bc4+ Be6 8.Bxe6+ Kxe6 (Short-Shirov, Dubai rapid 2002). I am not sure if White's compensation is objectively 100% sufficient in that line, but with the black king considerably misplaced there will always be significant practical chances. However, "Vass" at the Chesspublishing.com forum suggested 6...d5!?, which appears good for Black.
A deviation is 5.Nc3 (Topalov-Kramnik, Linares 1999) which is probably no better or worse than 5.d4 theoretically, as I haven't been able to find a way for White to avoid the transpositions to the critical line with ...g6 and ...c5, while Black's other responses tend to give White enough compensation. The Topalov and Short games are both analysed in my site's games and analysis section.
I think that the Cochrane Gambit isn't 100% sound with best play, but it is close enough to giving White full compensation to be fully viable at club level and maybe a fair way beyond that. Those who consider the knight sacrifice to be too risky could consider the line 4.Nf3 Nxe4 5.Nc3 Nxc3 6.dxc3 instead, while Michael Goeller wrote an extensive article on some approaches following 3.d4.
Another source on the Cochrane is Goeller's bibliography on the line.