I am a beginner playing white and it's my turn to move. What is the best possible move in response to this fork?
[title "Game position after fork:"] [fen "r1bqr1k1/ppp2p2/2n1pn1p/3p2p1/1b1P1N1B/2N1P3/PPPQBPPP/R3K2R w KQ - 0 1"]
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Disclaimer : in any case you are lost if your opponent plays reasonnably enough.
Assuming that you don't want to resign at move 10, my suggestion would be to continue with 10.Bg3 gf4 11.Bf4. Black's king is slightly weakened, you hit h6 and plan to continue with 0-0-0,f3,g4,h4,g5 to try and open his kingside as much as possible. In positions with opposite castles, material is not the single most important criteria, king security and initiative matter too.
If Black plays passively or conservatively, say 11...Kg7, then 12.f3 prevents Ne4, and with Bd3 and g4-g5 you may be able to muddy waters enough to induce later mistakes by Black.
If he plays actively and purposefully, with 11...Ne4 (or even 10...Ne4) 12.Qd3 e5!, not worrying about the Ph6 and intending ...Bf5, well, you hardly have anything to hope for...
The position is lost since you lose a piece for at most a pawn and no real compensation if black knows what to do.
For the best possible move in such situation you could consult a computer which would give you a move that it thinks is best based on its evaluation function. Likely it would still evaluate your position as -3 or so.
Anyway, if you are playing against a human, you don't want the objectively best move here, but the move (or line) that gives you the most practical chances or a move/line that gives your opponent more chances to go wrong. Some people call this "hope chess" which should normally be avoided, but in emergency situations could be your only hope.
Generally speaking this often means that in lost positions like this you:
For the position at hand, I believe the most practical chances would give you an attack on the black king. A few factors help you here: Black has weakened the pawn structure in front of his king, a large part of black's forces are concentrated on the queenside far away from the king. Your plan should be to move as many forces as possible to the kingside and to open a file or two for your rooks. Also acting quickly would be essential before black can make use of his extra piece.
Apart from winning a piece, black also has Ne4 which is very unpleasant because of the pin of the knight on c3; and e5 which frees black bishop on c8 and threatens to open the center against the uncastled white king. There is not much white can do to prevent these, or any attempt would basically mean passive play which is not what is needed here.
With this in mind you can think of how to achieve your goals and analyze concrete lines. Keep in mind that there is no (objective) right or wrong here, so I just give a few thoughts without saying that one line is better than the other.
Disclaimer: All of the above is for a position where your opponent still has to convert the advantage not for a position where you are lost because of a mating attack.
The position is completely lost. Computers don't evaluate swindle potential. I would try Nd3, because a half-open g-file has some value and the pawn (after ...gxh4) is quite weak. Meanwhile you hit the annoying bishop that is pinning your Knight. ...Bxc3; bc3 gh4; f3 (immediate e4 embarrassed by ...Nxe4) and hope your opponent makes a serious error.
There are three elements to chess. Space, time and forces. Since you are a man down, you can compensate with trying to gain more space and control and try to run their time down. Capturing the pawn grabs a pawn and messes up his space meaning it is extremely logical. 1.Bxg5 hxg5 2 Nh5 trying to gain initiative and rip open his position. White is also not so good because he isn't castled and the bishop will mess the pawn structure up. Try to complicate the position, create traps and avoid exchanges.