Definition of Soundness What exactly does it mean for a gambit to be unsound?  I see this term being used a lot but I have yet to come across a precise definition.  

Is it enough for white to lose the  advantage that comes with being the first player?  What if black is better with best play but not winning?

How is soundness defined if black is the one playing the gambit?
Adapted from here.

A gambit is said to be 'sound' if it is capable of resulting in an advantage against the opponent. There are three general criteria in which a gambit is often said to be sound:


Time gain: the player accepting the gambit must take time to obtain the sacrificed material and possibly must use more time to reorganize his pieces after the material is taken.
Generation of differential activity: often a player accepting a gambit will decentralize his pieces or pawns and his poorly placed pieces will allow the gambiteer to place his own pieces and pawns on squares that might otherwise have been inaccessible. In addition, bishops and rooks can become more active simply because the loss of pawns often gives rise to open files and diagonals. 
Generation of positional weaknesses: finally, accepting a gambit may lead to a compromised pawn structure, holes or other positional weaknesses.


An example of a sound gambit is the Scotch Gambit: 

[FEN ""]
1.e4 e5 2.Nf3 Nc6 3.d4 exd4 4.Bc4. 


Here Black can force White to sacrifice a pawn speculatively with 4...Bb4+, but White gets very good compensation for one pawn after 5.c3 dxc3 6.bxc3, or for two pawns after 6.0-0 inviting 6...cxb2 7.Bxb2, due to the development advantage and attacking chances against the black king. As a result, Black is often advised not to try to hold on to the extra pawn.

An example of an unsound gambit is the so-called Halloween Gambit: 

[fen ""]
1.e4 e5 2.Nf3 Nc6 3.Nc3 Nf6 4.Nxe5?! Nxe5 5.d4.


Here the investment (a knight for just one pawn) is too large for the moderate advantage of having a strong center.