Different between left-most and right-most derivation
Given a derivation tree for a word, you can "implement" it as a sequence of productions in many different ways. The leftmost derivation is the one in which you always expand the leftmost non-terminal. The rightmost derivation is the one in which you always expand the rightmost non-terminal.

For example, here are two parse trees borrowed from Wikipedia:


The leftmost derivation corresponding to the left parse tree is
$$
A \to A + A \to a + A \to a + A - A \to a + a - A \to a + a - a
$$
The rightmost derivation corresponding to the left parse tree is
$$
A \to A + A \to A + A - A \to A + A - a \to A + a - a \to a + a - a
$$
The leftmost derivation corresponding to the right parse tree is
$$
A \to A - A \to A + A - A \to a + A - A \to a + a - A \to a + a - a
$$
The rightmost derivation corresponding to the right parse tree is
$$
A \to A - A \to A - a \to A + A - a \to A + a - a \to a + a - a
$$