14 non-attacking bishops
We may consider the white-square bishops and the black-square bishops separately.
At most 7 bishops can be placed on white squares, namely, at most one bishop on each of the 7 white diagonals parallel to the h1-a8 diagonal. In fact, we can put bishops on the 7 white squares b1, d1, f1, h1, c8, e8, g8.
The solution for black-square bishops is just the mirror image of the solution for white-square bishops. At most one bishop can be placed on each of the 7 black diagonals parallel to the a1-h8 diagonal, and this can be achieved with bishops on a1, c1, e1, g1, b8, d8, f8.
32 non-attacking knights
We can put 32 knights on the board by putting knights on all the white squares or on all the black squares.
One way to see the we can't have more than 32 knights is to consider a knight's tour. If we number the squares from 1 to 64 in the order they are visited by the touring knight, then it's clear that our non-attacking knights can occupy at most one of the two squares 1 & 2, at most one of squares 3 & 4, at most one of squares 5 & 6, and so on.
But a knight's tour is somewhat difficult, and not really needed for this problem. All we really need is to divide the 64 squares of the chessboard into 32 pairs, each pair being separated by a knight's move. Since the 8 x 8 board can be cut up inti eight 2 x 4 boards, it will be enough to observe that the 2 x 4 board admits such a pairing (and therefore can hold at most 4 non-attacking knights), namely a1 & c2, a2 & c1, b1 & d2, b2 & d1.
Non-attacking knights on variant chessboards
It can be shown that, provided m,n > 2, the maximum possible number of knights on an m x n chessboard is ceiling(mn/2), that is, it's mn/2 if mn is even, (mn+1)/2 if mn is odd. This number can obviously be attained by putting all the knights on squares of one color. Proving that it's optimal is more work.
Let's say that an m x n chessboard has a "good pairing" if the set of squares can be partitioned into pairs (with one square left over if mn is odd), each pair being connected by a knight move. The existence of a good pairing follows from the existence of a knight's tour, but good pairings are easier to find than knight's tours. It will suffice to show that a good pairing exists whenever m,n > 2. In fact, it will suffice to show that a good pairing exists for 2 x 4, 3 x 3, 3 x 4, 3 x 5, 3 x 6, 5 x 5, and 5 x 6 chessboards, since every m x n chessboard with min(m,n) > 2 can be partitioned into rectangular pieces of those seven sizes, without using more than one piece with an odd number of squares. Constructing good pairings for those seven small boards is left to the reader. (The 3 x 4, 5 x 5, and 5 x 6 boards allow knight's tours.)
