1) Start in the leftmost column
2) If all queens are placed return true
3) Try all rows in the current column.  Do following for every tried row.
a) If the queen can be placed safely in this row then mark this [row,
    column] as part of the solution and recursively check if placing
    queen here leads to a solution.
b) If placing queen in [row, column] leads to a solution then return
    true.
c) If placing queen doesn't lead to a solution then umark this [row,
    column] (Backtrack) and go to step (a) to try other rows.
3) If all rows have been tried and nothing worked, return false to trigger
backtracking.

grid = SparseArray[{i_, j_} :> 8 (i - 1) + j, {8, 8}] // Normal;
init = ArrayRules@grid; (* saves which position corresponds to which of the 64 values *)

checkSafe[grid_, row_, col_] := Module[
{rules = ArrayRules@grid, leftdiag, partialRightDiag1,
partialRightDiag2, currentRow, rightdiag},

leftdiag =
Count[rules, PatternSequence[{i_,j_} /; (i + j == row + col) -> [BlackQueen]]];

currentRow = Count[grid[[row]], [BlackQueen]];

partialRightDiag1 = Rest@NestWhileList[# + 1 &, {row, col}, ! MemberQ[#, 8] &];
partialRightDiag2 = Rest@NestWhileList[# - 1 &, {row, col}, ! MemberQ[#, 1] &];

rightdiag = Count[Join[partialRightDiag1, partialRightDiag2] /. rules , [BlackQueen]];
{leftdiag, currentRow, rightdiag}]

NQueens[col_] := True /; col > 8;

NQueens[col_: 1] := Module[{column = col, temp},
Do[
temp = checkSafe[grid, row, column];
If[Total@temp == 0,
grid[[row, column]] = [BlackQueen];
If[NQueens[column = column + 1] == True, Break[]];
grid[[row, column]] = {row, column} /. init (*remove Queen*)];, {row, 8}]
]