All four knights are now developed, and the position is now completely symmetrical, leading to a somewhat drawish position.

White has many choices on how to reply, which can lead to various types of positions.

Spanish Four Knights

4. Bb5 is the Spanish four knights, resembling a Ruy Lopez-type position. White develops the bishop to a good square, preparing castling the next move. Black usually responds either symmetrically with 4...Bb4 or can play in a more aggressive manner with 4...Nd4.

Scotch Four Knights

4. d4 is the Scotch four knights, which usually transposes into the regular Scotch Game, Schmidt Variation, though a unique gambit line, the Belgrade Gambit, which arises after 4...exd4 5. Nd5 does exist, which isn't possible with the regular Scotch move order.

Italian Four Knights

4. Bc4 is the Italian four knights, which looks to be a logical square for the bishop, but this move is in fact an inaccuracy, as Black can play the center fork trick 4...Nxe4! in which Black has already equalized and has secured a good foothold in the center. Black can also play a bit quieter with 4...Bc5.

Glek Four Knights

4. g3!? is the Glek four knights, in which white looks to fianchetto the light square bishop, and play a bit more slowly, in the style of a King's Indian Attack.

Gunsberg Four Knights

4. a3!? is the interesting Gunsberg four knights, which is a useful waiting move. With this move, White prevents 4...Bb4, and has an improved version of the Center Fork Trick in response to 4...Bc5. As a result, black's best response is 4...d5 which is a reverse Scotch four knights.

h3 Four Knights

4. h3!? is a trendy line that was popularized by GM Anish Giri, which is a useful waiting move. White cuts down any possibilities of a Bg4 pin happening in the future, which generally leads to improved versions of reversed Black's options listed above.

Halloween Gambit

4. Nxe5?! is the Halloween Gambit, which is objectively dubious, but offers white many aggressive and practical chances, and can be quite effective in faster time controls.

For explanation of theory tables, see theory table and for notation, see algebraic notation..
1. e4 e5 2. Nf3 Nc6 3. Nc3 Nf6

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Wikipedia has related information at Four Knights Game

de Firmian, Nick. Modern Chess Openings, 15th edition. New York, NY: Random House, 2008.