You can take as an example the gradient of a vector field [math]\vec{\nabla} \vec{V}[/math], then the nabla operator [math]\vec{\nabla}[/math] is a covariant vector, or a tensor of the form (0,1) which takes one contravariant vector to a real number.

Update:
Since you updated your question, I'll try to write a more complete answer to that.

For example [math]x^{\mu}[/math](in 2D Minkowski space) has the components [math](a,b)[/math] then [math]x_{\mu}[/math] has the components [math](a,-b)[/math], of course you knew this. So the object [math]x_{\mu}[/math] is called a 1-form or a covariant vector. The [math]x^{\mu}[/math] object is called a contravariant vector. You can think of a contravariant vector just like a normal high school vector as a directed line, but the 1-form, or covariant vector is a set of numbered surfaces(or parallel planes) through witch the contravariant vector passes. The 1-form is just like the metric tensor [math]g_{\alpha \beta}[/math] another machine like thing(all such machines are tensors). It(the 1-form) converts a contravariant vector into a scalar function. So if you have the 1-form [math]\eta[/math] as a field and a contravariant vector [math]v[/math], as a field then [math]<v, \eta>[/math] gives you how many pieces of [math]\eta[/math] are pierced by [math]v[/math]. So every contravariant vector field has an associated 1-form field, and they both together give you the length(or how many planes got pierced by the vector, which is the same) of the vector.
For example a short list:

Euclidean/Cartesian coordinates:
Vector(contravariant components): [math](dx,dy,dz)[/math]
1-form(covariant components): [math](dx,dy,dz)[/math]
Length/magnitude: [math]dx^{2}+dy^{2}+dz^{2}[/math]

Spherical coordinates:
Vector: [math](dr,d\theta,d\phi)[/math]
1-form: [math](dr, r^{2}d\theta, r^{2}sin^{2}\theta d\phi)[/math]
Length/magnitude: [math]dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta d\phi^{2}[/math]

Quantum mechanics:
Vector: [math]|i>[/math](ket)
1-form: [math]<j|[/math](bra)
Length/magnitude: [math]<j|i>[/math]

SR:
Vector: [math](cdt,dr[/math]
1-form: [math](cdt, -dr)[/math]
Length/magnitude: [math]c^{2}dt^{2}-dr^{2}[/math]

And they all have different metrics of course.

But the whole thing is much more deeply connected, and you have to study a lot of mathematics to see the real thing ! I can't give you the actual reason, but it has to do with that you need the dual vector of a vector to get it's magnitude, it's not always just the sum of the squares, also vectors and dual vectors are not easily related. You have to find them for every case.