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.