Making sense out of covariance and contravariance I just read about co- and contravariant vectors and I am not sure that I got it right:
If we imagine that we have a n-dimensional manifold $M$ then a tangent space is spanned by the vectors $\partial_1,...,\partial_n.$ These guys transform from one coordinate system to another by 
$$ \frac{\partial}{\partial x^i} = \frac{\partial y^j}{\partial x^i} \frac{\partial}{\partial y^j}.$$
This transform is according to wikipedia called the covariant transform. Now, it is worth noticing that normally the covectors are the elements in the dual space. The basis vectors of the dual space are given by $dx_1,...,dx_n.$ They transform differently as 
$$dx^i = \frac{\partial x^i}{\partial y^j} dy^j.$$
Despite, although we transform covectors this transform is called contravariant. So somehow it seems as if the kind of transform does not fit to the kind of vector we are considering here and I don't see why this happens.
If you have any questions, please leave me a comment.
 A: Basically, vectors are called contravariant because their components transform oppositely to the basis vectors: if our change of coordinates is such that
$$ \frac{\partial}{\partial x^i} = \frac{\partial y^j}{\partial x^i} \frac{\partial}{\partial y^j}$$
then if we have a vector $\mathbf{V}$, its components $V^i_x$ in the $x$ coordinates are related to its components $V^i_y$ by
$$V^i_x = \frac{\partial x^i}{\partial y^j} V^j_y.$$
By the same logic, 1-forms are called covectors or covariant vectors because their components transform like the basis vectors, while the basis covectors transform like the components of vectors.
A: In modern mathematical terminology, a functor is called covariant when it preserves the direction of the morphisms, contravariant if it reverses it. For a given differentiable map between manifolds (of which a special case would be open sets within the same manifold), the derivative is a map between the associated tangent bundles. This defines a covariant functor. Pullback of differential forms (covector fields) is a map between the covector bundles in the opposite direction, and defines a contravariant functor. In other words, the association of the bundle of vector fields to a manifold is a covariant functor, of the bundle of 1-forms is a contravariant functor. A pretty confusing (apparent) discrepancy in terminology.
Spivak, in his comprehensive introduction to differential geometry vol 1, says about it (page 113)

Classical terminology used these same words [covariant and contravariant], and it just happens to have reversed this: a vector field is called a contravariant vector field, while a section of $T^\ast M$ is called a covariant vector field. And no one has had the gall or authority to reverse terminology so sanctified by years of usage. So it's very easy to remember which kind of vector field is covariant, and which contravariant - it's just the opposite of what it logically ought to be.

