A well known blunder in this opening is White's reply 3. fxe5??, which after 3... Qh4+, either loses material after 4. g3 Qxe4+ or severely exposes the White K to attack after 4. Ke2 Qxe4+ 5. Kf2 Bc5+.
The opening is named after Austrian master Ernst Falkbeer, who played it in an 1851 game against Adolf Anderssen.
Old Main Line: 3....e4
In this variation, Black's compensation for the sacrificed pawn primarily consists of his lead in development, coupled with the exposure of White's King. For example: 4.d3 Nf6 5.Dxe4 Nxe4 6.Nf3 Bc5 (Black aims at the weak f2 square). At Maehrisch-Ostrau 1923, Rudolf Spielmann vs. Siegbert Tarrasch continued: 7.Qe2 Bf5 (this was condemned by the Handbuch des Schachspiels because of White's next, though Black had already got into difficulties in the game Réti- Breyer, Budapest 1917 where 7....f5 8.Nfd2 Bf2+ 9.Kd1 Qxd5 10.Nc3 was played) 8.g4?! (in retrospect, prudent was 8. Nc3) 8... 0-0! 9. gxf5 Re8 and Black has a tremendous position, as he is bound to regain material and White's positional deficiencies will remain.
This line fell out of favor after WW2. Nimzovich Counter Gambit: 3...c6 This has become the most commonly played move after 3.exd5, with its most notable advocate being John Nunn. It is usually attributed to Aron Nimzowitsch, who successfully played it in Spielmann-Nimzowitsch, Munich 1906. However, Frank Marshall actually introduced the move to master play at Ostend 1905, defeating Richard Teichmann.
Although Black won both of those games, 3...c6 languished in obscurity for many years. White can respond with 4.Qe2, despite the drastic defeat inflicted on Alekhine by Paul Johner at Carlsbad 1911, although 4.Nc3 exf4 is much more common. Theory has not reached a definitive verdict, but the resulting positions are believed to offer Black more chances than 3...e4.
Jim West On Chess has a nice Blog article on the Falkbeer HERE