Here we have a $5\times 5$ board, with $25$ squares. Each domino covers two squares, and so reduces the number of uncovered squares by two. Thus, the number of uncovered squares is always odd, and so in particular is not zero. So we cannot cover the whole board.
Here we have a $6\times 6$ board, with $36$ squares. There are many different ways to cover it with $18$ non-overlapping dominos.
Here we have a $6\times 6$ board, with two black corner squares removed, leaving $18$ white squares and $16$ black ones. Each domino covers one black square and one white square, so the number of uncovered white squares is always two more than the number of uncovered black squares. Thus, the total number of uncovered squares can never be zero. So it is impossible to cover the board.