Uiua, 45 bytes

⊡⊗:"QRBNK":⍜≡⊡⊸↥:+@.[↥,,⊃(/↥=0|/=|=2/×|=1/↥)]⌵-°⍉⇡8_8⟜¤

Try it!

                                                     ⇡8_8   # coordinates of an 8x8 grid
                                                ⌵-°⍉       # the absolute difference of each x and y from the point
                       [   ⊃(                 )]            # push an array of boolean arrays with the coordinates that have:
                                         |=1/↥              #   a max distance of 1 (king)
                                    |=2/×                   #   a product of 2 (knight)
                                 |/=                        #   equal values (bishop) 
                             /↥=0                           #   a zero (rook)
                        ↥,,                                 #   a 1 in one of the previous two arrays (queen)
                    +@.                                     # convert into .s and /s
            ⍜≡⊡ ↥:                                     ⟜¤ # put the piece's char into each array
⊡⊗:"QRBNK":    ⊸                                          # pick the array with the corresponding index in "QRBNK"

This feels too long (mainly in the ⊃).

x86 machine code, 114 103 bytes

Input:

• AL - ASCII piece type (N/B/R/Q/K)
• BP - 0x88 index
• DI - Buffer

Output:

• DI - Last character + 1

Assembly source code:

    pusha
    mov di, offset disp_db - 3
    mov cx, 8
    mov bx, cx
    repne scasb
    mov dx, di
    sub dx, offset disp_db - 1
    pop si
    mov di, si

init_loop:
    mov ax, '.'
    mov cl, 8
    rep stosb
    mov al, 9
    mov cl, 7
    rep stosb
    inc ax
    stosb
    dec bx
    jnz init_loop

init_done:
    push di
    mov di, si

piece_loop:
    mov si, offset moves_knight - 2
    add si, dx
    lodsb
    add si, ax

vec_loop:
    lodsb
    cbw

sign_loop:
    mov bx, bp

dest_loop:
    add bx, ax
    test bl, 088h
    jnz vec_cont
    mov byte ptr [di + bx], '*'
    test dl, dl
    jp dest_loop

vec_cont:
    neg ax
    js sign_loop
    cmp al, 80h
    jnz vec_loop

done:
    popa
    mov [di + bp - 128], al
    ret

disp_db db "NBKRQ"

moves_knight db vec_knight - moves_knight - 1
moves_bishop db vec_bishop - moves_bishop - 1
moves_king   db vec_king   - moves_king   - 1
moves_rook   db vec_rook   - moves_rook   - 1
moves_queen  db vec_king   - moves_queen  - 1

vec_knight db 0Eh, 12h, 1Fh, 21h, 80h
vec_bishop db 0Fh, 11h, 80h
vec_king   db 0Fh, 11h
vec_rook   db 01h, 10h, 80h

Ruby, 125

anonymous function, prints to stdout.

RevB: golfed, but a bug fix brought it back to the same length as before.

->x,y,z{72.times{|i|v=y-i/9;u=x-i%=9
e=u*u+v*v
$><<(i<8?e<1?z:[r=0==v*u,b=u*u==v*v,b|r,3>e,5==e]["RBQKN"=~/#{z}/]??*:?.:$/)}}

Prints each of the 64 squares + 8 newlines = 72 characters individually. Relies on e, the square of the Euclidean distance between the current square and the given coordinates for checking king moves and knight moves (and also for printing the piece value z when the Euclidean distance is zero.)

Ungolfed in test program

f=->x,y,z{                         #x,y,character
  72.times{|i|                     #8 rows of (8 chars + newline) = 72 chars
    v=y-i/9;                       #vertical diff between y coord and current square
    u=x-i%=9                       #horizontal diff between x coord and current square. note i%=8
    e=u*u+v*v                      #square of euclidean distance
    $><<(                          #$> is stdout. send to it....
      i<8?                         #if not the newline column,
        e<1?z:                       #if the euclidean distance is 0, print the symbol for the piece, else
        [r=0==v*u,                   #TRUE if v or u is 0 (rook)
        b=u*u==v*v,                  #TRUE if abs(u)==abs(v) (bishop)
        b|r,                         #TRUE if either of the above are true (queen)
        3>e,                         #TRUE if e == 1 or 2 (king)
        5==e                         #TRUE if e == 5 (knight)
        ]["RBQKN"=~/#{z}/]??*:?.:    #select value from array corresponding to piece and print * or . accordingly
      $/                           #if newline column, print a newline
    )
  }
}

x=gets.to_i
y=gets.to_i
z=gets.chomp
f[x,y,z]

JavaScript (ES6), 137 130 bytes

f=
(x,y,p)=>`${1e8}`.repeat(8).replace(/./g,c=>+c?(i=x*x--,z=y,`
`):i+(j=z*z--)?`.*`[+!{K:i+j>3,N:i+j-5,R:i*j,B:b=i-j,Q:b*i*j}[p]]:p)
;

<div onchange=if(+x.value&&+y.value&&p.value)o.textContent=f(x.value,y.value,p.value)><input id=x type=number placeholder=X><input id=y type=number placeholder=Y><select id=p><option value=>Piece<option value=B>Bishop<option value=K>King<option value=N>Knight<option value=Q>Queen<option value=R>Rook</select><div><pre id=o></pre>

Note: Outputs one leading newline.

Explanation: Builds and scans though the string 100000000100000000100000000100000000100000000100000000100000000100000000. Each 1 indicates a new line where the relative coordinate x is decremented and the relative coordinate z is reset. Each 0 indicates a new square where the relative coordinate z is decremented. (y is reserved to reset z.) Then uses the fact that many of the moves can be categorised by the squares i and j of the relative coordinates (before they were decremented):

• (initial square) i + j == 0
• B: i == j
• K: i + j < 3
• N: i + j == 5
• Q: i == j || !i || !j
• R: !i || !j

Because - is shorter than == it's golfier to compute the negation and invert it later. (This lets me use the cute '.*'[] expression although it's actually the same length as a boring ?: expression.) * is also golfier than &&.

Java 8 lambda, 473 435 289 characters

Looks like this:

(R,C,f)->{String b="";for(int r=0,c,d,D;r<8;r++){for(c=0;c<8;c++){d=R-r<0?r-R:R-r;D=C-c<0?c-C:C-c;b+=R==r&&C==c?f:((f=='R'||f=='Q')&&(R==r||C==c))||((f=='B'||f=='Q')&&d==D)||(f=='K'&&((R==r&&D==1||C==c&&d==1)||(d==D&&d==1)))||(f=='N'&&(d==2&&D==1||d==1&&D==2))?"M":".";}b+="\n";}return b;}

Or ungolfed into a class:

public class Q89429 {

    static String chessMoves(int row, int column, char figure) {
        String board = "";

        for (int r = 0, c, deltaRow, deltaColumn; r < 8; r++) {
            for (c = 0; c < 8; c++) {
                deltaRow = row - r < 0 ? r - row : row - r;
                deltaColumn = column - c < 0 ? c - column : column - c;
                board += row == r && column == c ?
                        figure :
                        ((figure == 'R' || figure == 'Q') && (row == r || column == c))
                        || ((figure == 'B' || figure == 'Q') && deltaRow == deltaColumn)
                        || (figure == 'K' && (
                                (row == r && deltaColumn == 1 || column == c && deltaRow == 1)
                                || (deltaRow == deltaColumn && deltaRow == 1)))
                        || (figure == 'N' && (deltaRow == 2 && deltaColumn == 1 || deltaRow == 1 && deltaColumn == 2))
                        ? "M" : ".";
            }
            board += "\n";
        }

        return board;
    }
}

This is a TriFunction. It returns the chess field as a printable String. I wanted to use streams, and it looks quite good. It's like a 2D iteration, may be shorter without the streams. Switched to classic loops and saved a lot!

It can definitely be shortened by using a ternary, I will do that now.

Updates

Saved 38 characters by using a ternary.

Saved 146 characters by using good old loops. We should all abandon streams ;)

Ruby, 209 Bytes

->v,a,b{R=->x,y{x==a||y==b};B=->x,y{x-y==a-b||a-x==y-b};Q=->x,y{R.(x,y)||B.(x,y)};K=->x,y{((x-a).i+y-b).abs<2};N=->x,y{((a-x)*(b-y)).abs==2};(r=(0..7)).map{|x|r.map{|y|[x,y]==[a,b]?v: eval(v+".(x,y)??x:?o")}}}

This is a lambda that makes use of other lambdas, defined inside.