Part (a): You need to shade one square below the diagonal and one above, but once you've chosen the first square the position of the second is fixed. There are 6 squares below the diagonal, so 6 possibilities.
Part (b): Since the two squares must be on the diagonal, the question is how many different ways can you choose 2 things from 4. There are 4 ways of choosing the first, and for each of those there are 3 possibilities for the second (4*3 = 12), but we've counted each pair twice, so the answer is 4*3/2 = 6. (look up permutations and combinations)
Remaining part: This can only work if both squares are on a diagonal, and again once you've shaded one square you have no choice for the other one. There are 4 diagonal squares in the bottom half of the big square, so those are the only possibilities:
*--- ---- ---- ---*
---- -*-- --*- ----
---- --*- -*-- ----
---* ---- ---- *---
The last one could be done just by trying all the possibilities and making sure not to repeat any.