The way I see it, neither of you was aggressive in the opening -- you both tried to get rid of your pieces as quickly as possible! Being aggressive is getting many pieces in active positions quickly, not giving them away.
When we give advice, we have to take into account the level of the players. Both of you need to work on the most basic skill first: Piece safety. The first skill to learn is to recognize when undefended pieces can simply be captured, to always do that if your opponent gives you the chance, and to never give him the chance to just take a whole piece without repercussions. The moment one of you starts doing that, he'll blow the other off the board completely.
Let's show a number of positions where this was the case:
[FEN "rnbqkb1r/pppppppp/5n2/8/3PP3/8/PPP2PPP/RNBQKBNR b KQkq - 0 2"]
After 2.e4?, black can take a pawn for free, and he does.
[FEN "rnbqkb1r/pppppppp/8/8/2BP4/8/PPP2nPP/RNBQK1NR w KQkq - 0 4"]
Black gives away a full knight with 3...Nxf2. Why??
[FEN "rnbqkb1r/ppp1pppp/8/3B4/3P4/8/PPP2KPP/RNBQ2NR b kq - 0 5"]
White returns the favour immediately! Why give away a bishop with 5.Bxd5 when you could have just retreated it, say to d3?
[FEN "r1b1k2r/ppp2ppp/2n5/4q3/1b6/2P5/PP2Q1PP/RNB1K1NR w kq - 1 10"]
After 9...Nc6, black has done nothing to prevent white taking the bishop on b4! Had he played 9...Qxe2+ 10.Nxe2 Bc5 instead he wouldn't have lost a piece.
[FEN "r1b1k2r/ppp2ppp/2n5/4q3/1b6/2P2N2/PP2Q1PP/RNB1K2R b kq - 2 10"]
But white doesn't take it!
[FEN "r1b1k2r/ppp1qppp/2n5/8/1b6/2P2N2/PP2Q1PP/RNB1K2R w kq - 3 11"]
Black still ignores the threat to his bishop! He could still have played 10...Qxe2+ and then move the bishop.
[FEN "r4rk1/ppp1Bppp/2n5/2q5/6b1/P4N2/1P1NQ1PP/R3K2R b - - 2 15"]
White just played 15.Be7??, giving away his bishop. It's attacked twice (by the knight on c6 and the queen on c5), and only defended by the queen, so black could just have played 15...Qxe7 or 15...Nxe7 !
[FEN "r4rk1/ppp1Bppp/8/2q5/3n2b1/P4N2/1P1NQ1PP/R3K2R w - - 3 16"]
Instead, black insists on giving away a knight himself with 15...Nd4. If white simply answers 16.Bxc5 Nxe2 17.Kxe2, white has won a full knight.
[FEN "r4rk1/pppqBppp/8/8/6b1/P4N2/1P2Q1PP/R3K2R w - - 2 18"]
Now white is in some trouble, because he never bothered to get his king out of the center and a rook is threatening to come to the e-file, capturing the queen. There are two options: move the king away (it can't castle anymore, it moved to f2 in the opening) and defend the bishop with 18.Kf2 Rfe8 19.Rhe1. White will try to move his queen out of the pin next, say with 20.Qd1, so he's better. The other option is to take the offered rook with 18.Bxf8 Re8 19.Bb4 Rxe2+ 20.Kxe2, and white has two rooks and a bishop for the queen, which is good.
But the point is: if you see nothing of that, then you play Bxf8, because the rook is worth more than the bishop.
[FEN "4rrk1/ppp2ppp/8/2B5/q5b1/P4N2/1P2Q1PP/3RK2R w - - 6 20"]
But white didn't and we get to the end of the game, white is in trouble. But black for some reason moved the rook from a8 to e8, so that the one on f8 can still be captured.
After 19.Bxf8 Rxe2+ 20.Kxe2 Kxf8 white has two rooks for a queen and two pawns, his king is in the middle and his rooks aren't active. White is lost. But that means nothing as long as both players keep giving away pieces at the same rate as they've done so far, anything can still happen.
Edit: as AlwaysLearningNewStuff points out, in the final position white can block the attack on the queen with 19.Be3!, he's still a knight up for two pawns, and he can probably untangle his position with Kf2, Rhe1 et cetera. That's much better.