I'm trying to find a move that switches $2$ corner cubies of Rubik's Cube without disturbing the other $6$. And then a move which fixes the position of the back corner cubies and one front corner cubie.
I think there's something to do with commutator, but I just can't find what exactly. Can you help me to solve this, or just give some intuition.
On
The short answer is that the permutation group of the Rubick's cube is an subset of the Alternating group. That means that any permutations must be an even number of transpositions. You cannot just swap corners A and B leaving the rest unchanged. However, you can swap (AB)(AC) = (ACB)
If the top Layer of the cube is ABCD going around clockwise. You have figured out how to swap A for B, keeping the rest of the top layer unchanged, but making a mess out of the rest of the cube.
The top of your cube now BACD. Twist the top layer so that it is ACDB. Perform the inverse swap (the exact same moves you did to swap A for B in the exact reverse order). Not only will this swap A for C, it will unscramble the bottom of the cube.
If you don't care what happens to the edges, you could do $$ RU'RUR^2F^2RFR'F^2 $$ which interchanges the two top front corners and leaves the 6 other corners untouched. (By exhaustive computer search, there are no solutions shorter than this, in either the half-turn or quarter-turn metrics. Interestingly, this shortest solution only upsets 4 edges, even though the search allowed for arbitrary disruption of edges).
If all edges must remain untouched too, then the task is impossible, as Doug M explains.
A standard 3-cycle commutator will do that, this time leaving the edges alone:
$$ U'FDF' UFD'F' $$