General approach
Think of each of the 81 locations in the puzzle as containing a set of possible numbers. If the number for a particular location is given (for example, row 0 column 1 in the above Sudoku is 6), you can represent that as {6}, the "singleton" set containing only the number 6. If a location in the puzzle is blank, then any number might go there, and you can represent that as the set {1, 2, 3, 4, 5, 6, 7, 8, 9}. As you begin to solve the puzzle, you remove elements from these sets of possible numbers.

For example, 6 occurs in row 0, so it can't occur anyplace else in row 0. You can remove the 6 from every other set in row 0 that contains a 6: locations (0, 0), (0, 2), (0, 4), (0, 6), and (0, 8). The 6 is in column 1, so you can remove the 6 from every other row in that column: locations (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1) and (7, 1). The 6 is in the top left-hand "box" (rows 0 to 2 and columns 0 to 2), so you can remove the 6 from every other location in that box: locations (0, 0), (0, 2), (1, 0), (1, 1), (2, 1), and (2, 2).

A Sudoku puzzle is "solved" if every location in the array contains a singleton set (a set containing only one element).