Chess Opening Theory/1. e4/1...e5/2. Nf3/2...Nc6/3. Bb5/3...a6/4. Bxc6/4...dxc6/5. O-O

Castling reinvigorates the threat of Nxe5, because the tactic ...Qd4 to recover the pawn will fail to Nf3 Qxe4 Re1! pinning Black's queen to the king. Therefore, Black must prevent Nxe5.

5...Bg4 accomplishes this by pinning the knight. Because White has castled and Black has not, this bishop can be difficult to dislodge. 6. h3 h5 is the Alapin gambit: if White takes then they can face trouble on the open h-file, e.g. 7. hxg4 hxg4 8. Nxe5 (h-pawn threatened to take knight) Qh4! is mate in 3 (9. f4 g3 10. Qh5 Rxh5 11. Re1 Qh1#).

Guarding the pawn is the other approach:

5...f6, the Gligorić variation, is the mainline and the most common way to defend the pawn. It may look ugly, and it prevents Black's knight from coming to f3, but it is less dangerous now White's light square bishop is off the board and quite often Black ends up playing it in other lines anyway. If White presses the issue of e5 with 6. d4, then 6...exd4 7. Nxd4 c5 8. Nb3 Qxd1 9. Rxd1 leads to a queen trade.

5...Bd6, the King's bishop variation, is an alternative. This avoids a queen trade after 6. d4 exd4 7. Qxd4 though Black still must play 8...f6 anyway, as White's queen threatens g7 and 8...Nf6?? runs into 9. e5! forking the bishop and knight.

5...Qd6 is the Bronstein variation. Black may wish to play ...Be6 or ...Bg4 and ...O-O-O, forming a battery on the d-file.

This is sometimes called the Barendregt variation for Johan Barendregt (1924―1982).

Wikipedia has related information at Ruy Lopez, Exchange Variation