Tell me the moves

Python, 217 212 220 217 213 characters

Tied the 213-byte Mathematica solution

R=range(8)
def f((p,x,y)):
 for a in R:
    for b in R:
     A,B=abs(a-ord(x)+97),abs(b-ord(y)+49);C=max(A,B);r=(A+B==3and C<3,C<2,A*B<1,A==B,0)
     if(r['NKRBQ'.index(p)],any(r[1:]))[p=='Q']*C:print p+chr(a+97)+chr(b+49)

I started off by generating all valid moves but that grew too large so the approach is quite similar to the Mathematica one.

>>> f("Nd4")
Nb3
Nb5
Nc2
Nc6
Ne2
Ne6
Nf3
Nf5
>>> f("Qa1")
Qa2
Qa3
Qa4
Qa5
Qa6
Qa7
Qa8
Qb1
Qb2
Qc1
Qc3
Qd1
Qd4
Qe1
Qe5
Qf1
Qf6
Qg1
Qg7
Qh1
Qh8

Mathematica, 278 272 264 260 215 213 characters

f=(FromCharacterCode@Flatten[Table[c=Abs[#2-x];d=Abs[#3-y];b=c==d;r=#2==x||#3==y;If[Switch[#-75,0,c~Max~d<2,-9,b,7,r,6,b||r,3,!r&&c+d==3],{p,x,y},##&[]],{x,97,104},{y,49,56}]&@@ToCharacterCode@#,1]~DeleteCases~#)&

Ungolfed version:

f[pos_] := (
  {piece, u, v} = ToCharacterCode@pos;
  board = Flatten[Table[{piece, i + 96, j + 48}, {i, 8}, {j, 8}], 1];
  DeleteCases[
    FromCharacterCode[
      Cases[board, {_, x_, y_} /; Switch[p,
        75, (* K *)
        ChessboardDistance[{x, y}, {u, v}] < 2,
        66, (* B *)
        Abs[u - x] == Abs[v - y],
        82, (* R *)
        u == x || v == y,
        81, (* Q *)
        Abs[u - x] == Abs[v - y] || u == x || v == y,
        78, (* N *)
        u != x && v != y && ManhattanDistance[{x, y}, {u, v}] == 3
        ]
      ]
    ], 
    pos (* remove the input position *)
  ]
)&

Example usage:

f["Nd4"]
> {"Nb3", "Nb5", "Nc2", "Nc6", "Ne2", "Ne6", "Nf3", "Nf5"}

The ungolfed version creates a full board, and then selects correct positions with Cases, whereas the golfed version drops invalid moves immediately in the Table command by issuing ##&[], which simply vanishes.

Haskell 225 220 208 205 200 182

f=fromEnum
m[p,a,b]=[[p,c,r]|c<-"abcdefgh",r<-"12345678",let{s=abs$f a-f c;t=abs$f b-f r;g"K"=s<2&&t<2;g"Q"=g"B"||g"R";g"N"=s+t==3&&(s-t)^2<2;g"B"=s==t;g"R"=s<1||t<1}in s+t>0&&g[p]]

It's going to be difficult to touch Mathematica when that has chess moves built in :rollseyes: (well played m.buettner) I take it all back. Beating Mathematica by 31!

Latest edit: replaced case with a function, inlined filter into comprehension, to beat the entry in R ;)

usage:

ghci> m "Nd4"
["Nb3","Nb5","Nc2","Nc6","Ne2","Ne6","Nf3","Nf5"]

Ungolfed (corresponds to 208 char version before 'u' was inlined):

f=fromEnum -- fromEnum is 'ord' but for all enum types,
           -- and it's in the prelude, so you don't need an extra import.
u piece dx dy= -- piece is the character eg 'K', dx/dy are absolute so >=0.
  dx+dy > 0 && -- the piece must move.
  case piece of
    'K'->dx<2&&dy<2         -- '<2' works because we already checked dx+dy>0
    'Q'->dx<1||dy<1||dx==dy -- rook or bishop move. see below.
    'N'->dx+dy == 3 &&      -- either 2+1 or 3+0. Exclude the other...
         (dx-dy)^2 < 2      -- 1^2 or 3^2, so valid move is '<2', ie '==1'
    'B'->dx==dy             -- if dx==dy, dx/=0 - we checked that. 
                            -- other moves with dx==dy are along diagonal
    _->dx<1||dy<1           -- use _ not 'R' to save space, default case is
                            -- the rook. '<1' saves chars over '==0'.
                            -- Again, dx==dy==0 edge case is excluded.
m[piece,file,rank]=       -- the move for a piece. 'parse' by pattern match.
 filter(                    -- filter...
  \[_,newfile,newrank]->    -- ...each possible move...
    u piece                 -- ...by, as everyone noticed, converting char..
      (abs$f file-f newfile) -- differences to absolute dx, dy differences,..
      (abs$f rank-f newrank)) -- and then using special routines per piece.
    [[piece,newfile, newrank] -- the output format requires these 3 things.
      |newfile<-"abcdefgh",newrank<-"12345678"] -- and this just generates moves.