In summary, covariant and contravariant vectors have different uses and transform in opposite ways under a change of coordinates. Covariant vectors are used to approximate scalar fields while contravariant vectors are used to approximate parametrized curves.
That's a question either for the relativity or differential geometry section rather than quantum physics, although you sometimes see it Quantum Field theory.
Can someone explain to me what is the difference between covariant and contravariant vectors ? Thank You
Given Bill's pointer to a web page talking about the difference, I'm not sure if it's appropriate to add anything, but what I find most useful is not the mathematics for how the two kinds of vectors transform, but what they are good for. The typical use for a regular vector is as a "tangent" or "local approximation" to a parametrized curve--for example, a velocity vector [itex]\vec{v}[/itex] describes how a position as a function of time is behaving locally. The typical use for a covector is a "local approximation" to a scalar field (a scalar field is a real-valued function of location, such as altitude or temperature on the Earth at a given time). In vector calculus in Cartesian coordinates, you would use [itex]\nabla T[/itex] to describe how the scalar field [itex]T[/itex] changes locally. The components of the two types of vectors transform in opposite ways under a change of coordinates.