I'm a bit confused about the various geometric concepts of vectors.
I'm mainly trying to understand if we can classify any vector into one of two categories.The first category would be free vectors, that is an element who is specified only by magnitude and direction, regardless of where it starts and ends ( regardless of the position ).
The second category would be bound vectors, that is, an element who is specified not only by magnitude and direction, but by where it starts and ends ( by its position ).
Should i be happy with this two categories when trying to understand and classify any kind of vector, or there is more to it ?
I went for a quick search in the wikipedia and found some kinds of vector :
Euclidean Vector, Displacement vector ( physics ), Position vector, Gradient Vector ( a vector that arises in vector calculus ) and "the vectors that define a Vector Field".
Would the position vector, gradient vector be examples of "free vectors" and would the Euclidean vector, Sisplacement vector and the vectors that define a Vector Field examples of "bound vectors" ?
Can i classify each kind of vector i mentioned in either bound or free vector categories ? Can i mantain the geometrical concept of vector as being either bound or free in my mind? Am i missing anything ?
This is a tricky question, because there are a number of slightly different meanings of "vector" in mathematics, so the correct answer depends on which area you're working in.
However, at the level of sophistication where we're talking about "magnitude and direction" as a possible definition of vectors, the answer is unambiguously that a vector is always what you call a "free" vector. It is important for the application of vectors to geometry that the vector from $(0,0)$ to $(1,2)$ is the same vector as the one that takes from $(42,1000)$ to $(43,1002)$. Of course you sometimes need to wrap up the combination of a starting point and a vector into a single mathematical object, and it's fine to do so -- just don't call the combined object a "vector", because that will confuse people. It's just a pair of a point and a vector (or, equivalently in Euclidean space, an ordered pair of a starting point and an endpoint).
In linear algebra we meet the concept of an abstract vector space, which is generalized to contexts where "magnitude and direction" make little sense. In this abstract sense a vector is any member of something that satisfies the vector space axioms. Note that the set of what you call "bound" vectors doesn't satisfy these axioms (with any reasonable definition of what vector addition and scalar multiplication should mean), so they are not even abstract vectors.
By the way, since your examples sound a bit like you might be coming at this from a physics viewpoint, note that physicists usually don't call the elements of arbitrary abstract vector spaces "vectors" at all. They have other uses for the word.
What about vector fields, you ask? A vector field is not one vector -- it is a function that assigns some vector to each point of some space. However, the vectors that are values of a vector field are ordinary ("free") vectors -- in particular they don't know at which point of the vector field we got them out of. This is important too, because in many applications we're interested in whether the vector field has the same (or approximately the same) values at two different points, and we wouldn't be able to speak of that if the identity of one of the vector field's values depended intrinsically on which point it was a value at.
(An aside: It is not uncommon in physics jargon to speak of a vector field as simply "a vector", especially when speaking about equations that are supposed to hold at every point in the space. Strictly speaking that's an abuse of terminology, but it seems to work well for them. It doesn't seem to have anything to do with the incidental fact that the set of all possible vector fields constitute an abstract vector space in the mathematical sense).
Unfortunately, there's an exception to the above, namely when the vector field doesn't live in the usual Euclidean space but in a more general manifold, such as a curved surface in space. In that case each point on the surface has its own private space of tangent vectors, and a vector field then selects for each point a particular vector in its private tangent space. Then the vectors do know which point's tangent space they belong to, and we have to take special measures if we want to compare the value of the vector field at two different points (for example, to differentiate it).
As a simple example of this, consider the surface of the earth, and a velocity field on it. The velocity vectors need to stay within the surface, so at a flat piece of ground the velocity can be north, sough, east, and west or some combination of them -- but not up an down. However, on the side of a steep hill, "due north" might not be available as a direction at all, because that points into the ground, and similarly "south" points into free air. So each point on the surface has its own space of possible tangent vectors.
Note, however that the tangent vectors of a manifold don't behave quite like you describe "bound" vectors. In your description, a "bound" vector is supposed to have a start point and an end point, but a tangent vector is tied to one point on the surface, with no distinction between a start point and an end point. Again, a velocity vector is a good example -- if a particle is currently at a particular point and moving due north at $10^{-9}$ times the speed of light, it makes little sense to attach a separate endpoint to its velocity vector.
Confused yet?