There are a couple problems with the King's Gambit.
It has a few "refutations" floating about, but they are really not so terribly awful for white.
The biggest problem is the number of different defenses black can play. (Many of which don't give white the attacking game he/she is hoping for.)
1.e4 e5 2.f4
A. 2. ... d5 3.exd5 e4 4.d3 Nf6 5.dxe4 Nxe4 6.Nf3 Bc5 7.Qe2 Bf5 8.Nc3 Qe7 9.Be3 is actually analyzed out to a better ending for white.
This line really puts the whole Falkbeer with 2. ... d5 in bad shape.
B. 2. ... Bc5 is playable for black. 3.Nf3 d6 4. Nc3 (This is a nice move because it keeps the option of Bb5 open for white.) Nf6 5.Bc4 is the typical King's Gambit Declined. White plays d3, sometimes Na4 to win the two bishops, and sometimes white can rush in a kingside attack with f5, h3, g4, etc.
C. 2. ... exf4 (The main accepted variation)
C.1. 3.Bc4 Nf6 (3. ... Qh4+ 4.Kf1 is fine for white. White will gain time with Nf3 attacking the queen later.) 4.Nc3 (4.d3/4.Qe2 are interesting as well) c6 or Bb4 are the main lines for black.
Here is where black has so many choices
3. ... g5 is the old main line
3. ... d6 is the Fischer variation
3. ... d5 is the Modern variation
3. ... Nf6 is also interesting
3. ... d5 4.exd5 Nf6 5.Nc3 is interesting Nxd5 6.Nxd5 Qxd5 7.d4 is one variation. I don't think 5.Nc3 is the main line, but it can be used here and against
3. ... Nf6 4.Nc3 d5 transposes
3. ... d6 4.d4 g5 5.h4 g4 6.Ng1 Qf6 7.Nc3 c6 8.Nge2 Bh6 9.g3 is one variation Black is very weak on the f-file. If 9. ... f3, white gets control of the f4 square.
3. ... g5 4.Bc4 (4.h4 is the usual move, but it has its problems theoretically. g4 5.Ne5 ... 5.Ng5 h6 6.Nxf7 is the wild Algaier Gambit ...
Nf6 6.d4 d6 7.Nd3 Nxe4 8.Bxf4 is the Kiesertsky Gambit. White gets good play down the f file. I remember black had a good way to continue this, the last time I looked at it. White was in rough shape.)
Back to 4.Bc4 Bg7 (4. ... g4 5.0-0 gxf3 6.Qxf3 is the Muzio Gambit. It is also a very wild piece sacrifice. It is very fun to analyze but the books have it as a draw. The worst part is that hardly anyone plays 4. ... g4) 5.d4 (or 5.h4 h6) d6 is playable and perhaps slightly better for black.
Overall the King's Gambit is a good tactical opening.
Black can take a lot of the fun out of it if he knows what he's doing, though.