The Urusov Gambit can be reached via various move-orders. The most popular is 1.e4 e5 2.Bc4 Nf6 3.d4 exd4 4.Nf3, but fans of the Scotch and Göring Gambits can enter the line via 1.e4 e5 2.d4 exd4 3.Bc4 Nf6 4.Nf3, or 1.e4 e5 2.d4 exd4 3.Nf3 Nf6 4.Bc4. In the third case, Tim Harding has mentioned that "White usually plays 4.e5 instead of 4.Bc4", but in casual games I've quite often got into the Urusov Gambit using that particular move-order, unable to resist the opportunity to enter these lines.
White offers a sacrifice of the pawn on e4, with the idea being that after 4...Nxe4 5.Qxd4, White's queen takes up a commanding central position, and can go to an aggressive post at h4, bearing down on Black's kingside. White appears to get sufficient compensation for the pawn in all lines. For a thorough coverage of the Urusov I refer readers to Michael Goeller's Urusov Gambit site. There is also a good analysis of the accepted lines of the gambit in Danish Dynamite.
In this position White is threatening Bc4-d3, putting more pressure on Black's kingside, and if Black plays ...h7-h6 then a strong Bg5xh6 sacrifice follows. Black is already in serious trouble in the position in the above diagram, which is why Black is advised not to castle too early in the Urusov Gambit, and cannot just trot out the standard defensive moves associated with the Hungarian Defence (1.e4 e5 2.Nf3 Nc6 3.Bc4 Be7).
Minor 4th moves for Black
Black can decline the gambit in various ways. 4...c5 is not as bad as it looks, although it leaves a hole on d5 which White can exploit with 5.0-0 intending to meet 5...Nf6 with either 6.Ng5 or 6.Re1 followed by 7.c3. 4...Bc5 should be met by 5.e5 d5 6.exf6 dxc4 7.Qe2+, which leads to Max Lange-esque complications which are quite favourable for White. 4...d6 should be met by Michael Goeller's suggestion of 5.0-0, intending to recapture on d4 in most cases, leading into lines of the Antonishin Variation of the Philidor Defence, which offer White a slight space advantage and some kingside attacking chances.
4...Bb4+ and 4...d5
4...Bb4+ is quite a strong move, since I don't think Goeller's suggestions after 5.c3 dxc3 6.bxc3 d5! (counterattacking against the bishop on c4) 7.Qa4+ Qd7 work out very well for White. Therefore I prefer the bold sacrifice of a second pawn with 6.0-0. Play can then transpose into Scotch Gambit lines following 6...cxb2 7.Bxb2 Nc6, though these are dangerous for Black, but Black also has the independent option of 7...0-0 8.e5 Ne4. My feeling is that White probably gets two pawns' worth of compensation, but 4...Bb4+ is a good winning try. 4...d5 is sometimes recommended as a way to equalise, but I agree with Goeller that White can probably get an edge with 5.exd5 Bb4+ 6.Kf1!?, which generally wins a pawn, though at the cost of a misplaced king. I think that White has similarly good chances of an edge with 6.c3 Qe7+ 7.Be2 dxc3 8.Nxc3 (instead of the more popular 8.bxc3) and the resulting positions are generally easier to play, so this is my main recommendation for White here.
This sensible move transposes into Two Knights Defence lines, which I cover via the Scotch Gambit.
4...Nxe4 5.Qxd4 sidelines
This is the main starting position for the Urusov Gambit. Black should retreat the knight to f6 here, helping to guard the kingside, since 5...Nd6?! 6.0-0 leaves Black under considerable pressure down the e-file and on the kingside. White's main idea is then to pin the f6-knight with Bg5, but White should play 6.Nc3 first, in order to gain control over the d5-square, since 6.Bg5?! is met by 6...Nc6 7.Qh4 d5, which gives Black a significant share of the initiative. Black's main independent alternative is to kick the white queen immediately with 6...Nc6 7.Qh4, and now 7...Bb4 is quite strong, discouraging White from castling queenside and carrying out the normal kingside attacking plans, and then preparing ...h7-h6, challenging White to sacrifice the dark-squared bishop on h6 under less favourable circumstances than usual. White should take a closer look at the interesting 8.Bd2, parrying the pin, as Goeller's main line with 8.0-0 Bxc3 9.bxc3 is not looking too strong.
4...Nxe4 5.Qxd4 Nf6 6.Nc3 Be7 7.Bg5
This is the main line of the Urusov Gambit Accepted, and despite Black's absence of weaknesses, White appears able to get sufficient attacking chances for the pawn. The key variations are 7...Nc6 8.Qh4 d5 9.Nxd5, 7...Nc6 8.Qh4 d6 9.0-0-0 Be6 10.Rhe1! Bxc4 11.Qxc4 0-0, and 7...c6 8.0-0-0 d5 9.Rhe1 (or 9.Qh4) 9...Be6 10.Qh4 Nbd7. The resulting positions offer good winning chances for both sides, while I think White has few problems getting enough compensation for the pawn.