TITLE: *understanding* covariance vs. contravariance & raising / lowering
QUESTION [16 upvotes]: There are lots of articles, all over the place about the distinction between covariant vectors and contravariant vectors - after struggling through many of them, I think I'm starting to get the idea.  I'm wondering if some-one/people can help me understand the meaning / significance of it.
My background is physics, with no training in 'abstract algebra' per se.  The language I use below is basically as technical as I understand (sorry...).

My understanding:
Contravariant vectors (like normal displacement/velocity/etc vectors---with upper indices: $\vec{v} = v^i \, \bf{e}_i$, on a basis $\vec{\bf{e}} = \bf{e}_i$) transform like,
$$v^{\prime j} = \frac{dx^{\prime j}}{dx^i} v^i.$$
Covariant vectors (like gradient vectors–with lower indices: $\vec{w} = w_i \,  \bf{e}^i $, on a basis $\vec{\bf{e}} = \bf{e}^i$) transform like,
$$w_{\prime j} = \frac{dx^i}{dx^{\prime j}} w_i.$$
We also have a metric $g_{ij}$ or $g^{ij}$ which can transform a contravariant vector to a covariant vector.

Questions:
If these vectors exist in the same 'space' (vector space?), and we want to make them interact---i.e. take the dot-product between them (which requires one to be upper-index and the other lower-index), then why are they being expressed with difference bases (basises?)?---doesn't that mean they're in different reference frames?
What is the meaning behind changing a vector from covariant to contravariant?  The components change in some way, but the 'meaning' of the vector is supposed to stay the same, right?
Does something being a contravariant vector simply mean it is being defined with respect to a basis of tangent vectors; while a covariant vector is one in reference to a basis of normal vectors?
[this is my interpretation of the first figure of http://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors ]

REPLY [2 votes]: i would like t add a further view on this question (complemetary to the already nice answers)
Both co-variant and contra-variant vectors are just vectors (or more generaly 1-order tensors).
Furthermore they are vectors which relate to the same underlying space (for example euclidean space or generally, a manifold)
Furthermore they relate to the same space but in different but dual ways (as such they have different, but dual, transformation laws, as already stated)
Co-variant vectors are part of what is called the tangent space, which for an euclidean space coincides or is isomorphic to the space itself. And contra-variant vectors are part of the dual tangent space, called co-tangent space (as already noted in other answers) and which for an euclidean space also coincides or is also isomorphic to the space itself.
These spaces (and their vectors) are dual, in the algebraic sense, related through the norm (inner product) of the space. Plus are isomorphic to each other regardless if they are isomorphic to the underlying manifold itself (what is called raising and lowering indices)
A question is what these (associated) spaces represent, how are they related and what is the intuition behind their use?
Historicaly tensors and tensor analysis was initiated as a by-product of the theory of invariants. A way was needed to express invariant quantities under a change of representation (or change of underlying basis). Thus tensors were used. tensors represent quantities which transform under a change of representation in such ways as to make various quantities expressed in terms of them invariant.
Note, the terminology association with co-variant/contra-variant indices is largely an convention, any consistent convention will do.
This also gives the (intuitive) relation between co-variant tensors (vectors) and contra-variant tensors (vectors).  When a co-variant vector (components) transform in one way, for example by a scaling factor $s$. The (associated) contra-variant vector (components) will have to transform by the inverse scaling factor $1/s$ in order for invariant quantities (for example an inner product $a^ib_i$) to remain invariant.