We learned that when changing coordinate system from $u^i$ to $u'^i$, a contravariant vector transforms like this (using the Einstein-convencion):
$v'^i = \frac{\partial x'^i}{\partial x^j}v^j$,
And a covariant like this:
$\omega'_i = \frac{\partial x^i}{\partial x'^j}\omega_j$.
I, however, have problems with understanding this, since I lack an intuitive picture about the event... What does the $\frac{\partial x'^i}{\partial x^j}$ derivate mean, and how does this transform rule work?
Thank you for the help in advance. This lack of understanding pretty much causes difficiulties in understanding the whole "tensory stuff", beginning with the differences between contra- and covariance...
In physics (and in mathematics, in differential geometry) you are working with so called manifolds $M$, which you can think of as smooth $k-$ dimensional subsets of some $\mathbb{R}^n$. A vector (field) along $M$ can be thought of as an assignment of a vector $X(p)$ in each point $p$ of $M$ which is tangent to $M$. A differential form is an assignment of a linear map from these tangent vectors to $\mathbb{R}$ in each point $p\in M$.
The classical description of such a manifold is by local coordiante systems, i.e. by mappings $x: U(p)\cap M\rightarrow \mathbb{R}^k$ of some neighbourhood of a given point $p$ to some Euclidean space. (You smmothly bend the nonlinear object to some linear space). If you have such a coordinate system then you also have local coordinate representations of vectors and forms (these are given by applying the differential of $x$ in $p$ to $X(p)$ resulting in a vector representation of $X$ in the target $\mathbb{R}^k$)
Now the original vector (field) $X$ is a physical or geometrical quantity (as is a form), so the behaviour of this object must not depend on the choice of a coordinate system. In other words, if you have two coordinate systems $x, x^\prime :U(p)\rightarrow \mathbb{R}^k$ you will end up with some conditions on the representation of a vector field $X$ with regard to $x$ or $x^\prime$.
If you have two such coordinate systems you also get a transformation from one coordinate system to the other one, by looking at $\Phi= x^\prime \circ x^{-1} :\mathbb{R}^k\rightarrow \mathbb{R}^k $
It turns out (by calculation, basically using the chain rule) that the required invariance condition can be expressed by looking at how $\Phi$ transforms vectors and forms. In order for some expression in one coordinate system to represent a vector field (on $M$) it has to transform in a certain way (the way you have written down) for each change of coordinate system, and a similar rule applies to differential forms.
That's about it.