What does "dimension" mean in physics? Still I don't get what's the difference between dimension and quantity. Are they same or they are different?
 A: The dimension, or physical dimensionality, is a descriptor of a quantity. Thus "the distance between New York and Los Angeles" is a quantity, which has dimensions of length.
A bit more precisely, you can see nouns like length, velocity, time, and mass as describing abstract quantities, i.e. "a velocity $v$" specifies that $v$ is a velocity but nothing else. Physical dimensionality is, at its heart, an equivalence relationship: in essence, you are always saying "quantity $A$ and quantity $B$ have the same physical dimensionality". Thus all lengths have the same physical dimensionality, and "the distance between New York and Los Angeles" has the same physical dimensionality as any length, which we just describe as 'length' in the abstract. 
Note in particular that "length" (no article), in the abstract, is a physical dimension, whereas "a length" refers to some arbitrary quantity. The lines between those two are pretty blurry - because it really doesn't matter all that much, as there isn't much harm in the ambiguity.
I hope that makes it clearer instead of more confusing.
