Fig 1. Position after 4.Bc4 Nxe4 5.Nxe4 d5.
What is the best way for white to give back the material?
In this position, white has a number of options, only one of which does not hand the opening advantage to black. The main moves are 6. Bxd5, 6.Bb5?, and 6.Bd3. Other moves are, of course, possible, but these are the most commonly played ones. Let's consider each of them.
First White Option: 6.Bxd5
This is usually followed up by the following sequence: 6...Qxd5 7.Nc3 Qd8 (fig 2).
Fig 2. Position after 6.Bxd5 Qxd5 7.Nc3 Qd8
The computer engines prefer 7...Qd7, but ...Qd8 is a more human move. It allows the two bishops to immediately develop and is just as good. Let's evaluate the position. Black has better central control due to his pawn on e5 and has the bishop pair, but white has no weaknesses. Thus, black is slightly better. A good black plan is to castle kingside and play for central control, space, and kingside pressure with ...Bd6, ...f5 and ...Bg4 if white does not play h3 to prevent it. The weakening of light squares on the a2-g8 diagonal after ...f5 are not cause for concern as white lacks a light square bishop.
Second White Option: 6.Bb5?
This is the third most popular move in my database (after Bxd5 and Bd3), but it is nearly losing--as long as black knows the refutation after 6.Bb5? dxe4 7.Nxe5 when he is threatening 8.Nxc6 bxc6 9. Bxc6+ followed by Bxa8. The most obvious (and third most popular response) is 7...Bd7?, which breaks the pin. After 8.Bxc6 black's pawns will be doubled, and white is better, but black does not have to settle for this. Another possibility, which is good but not the best, is 7...Qd5, the most popular move. After 8.Bxc4 bxc4 black has some pressure against white's position, but it is not much. 7...Qg5! (fig 3) is the refutation of this line and is almost completely winning. In my database it scores 83.3% for black!
Fig 3. Position after 6.Bb5? dxe4 7.Nxe5 Qg5!
The first thing to note is that it 7...Qg5, like 7...Qd5, neutralizes the 8.Nxc6 line that will end in a fork of the king and rook after 8...bxc6. With the queen on d5 or g5, black plays 8...Qxb5 instead of 8...bxc6 and after 9.Ne4 Qg5, white will be unable to castle kingside due to the threat of Bh3. White is worse, but gets to play on. If white plays 8.Bxc6+ bxc6 9.Nxc6, he wins a pawn (which is not possible in the 7...Qd5 line), but loses the game! Black should ignore the Nc6 and respond with 9...Qxg2 10.Rf1 Bg4, at which point white can resign. For a grandmaster slap down of an amateur that ends with 15...Qff2#--a rare bird in chess notation--see Zerzhalova, S (1953) - Afromeev, V (2628) 0-1.
Third White Option: 6.Bd3.
This move is the only one that does not give black an opening advantage. In fact for a long time it was believed that white emerges with a slight advantage after 6...dxe4 7.Bxe4 Bd6 8.0-0 0-0 9.Re1. You can even find a few examples of this on a high level such as the following blitz encounter: Ivanchuk, V (2739) - Karjakin, S (2723) 0.5-0.5.
But Larry Kaufman claims in The Kaufman Repertoire for Black and White (2012), that the novelty 6...Nb4 (fig 4) equalizes.
Fig 4. 6...Nb4, an Equalizer?
This move is based on the idea that black will get back his piece even if Ne4 moves away because 7...e4 introduces a second fork. So, 7.Ng3 (or Nc3 or even Neg5), e4 8.Bxe4 dxe4 9.Nxe4. White is temporarily up a pawn here, but after 9...Bf5, he has problems. The white knight can't move away because of 10...Nxc2+. There are several possible lines here (and I'll leave it to you to figure them out), but it appears that they all give black at least equality. Only a handful of games have been played with 6...Nb4, so it is possible that improvements will be found for white, but this looks promising for black.
So, there you have it: the easy guide to the 4.Bc4?! Four Knights! There are some other reasonable options that I have not covered such as the 4...Nxe4 5.0-0, which is a decent try for white. Try analyzing it yourself or go here for a brief discussion of it and more complete coverage of all of these lines.