English names for vector beginning and end I've done some research, but since English is not my native language, I'm struggling to find an answer to this:
Given a vector, what do you call its beginning and end points?
The best I've found so far is the word "base" for the beginning point of a vector, but I have no clue if that's correct. The best I've got for the end point is the "tip" of the vector, although the same applies as before.
 A: They are sometimes called the initial and terminal points. The initial point is the point at which starts and the terminal point is the point at which it ends.
A: The Wikipedia article Euclidean vector says that when you construct a vector, called $\overrightarrow{AB}$, from two points $A$ and $B$ in Euclidean space, then $A$ is called initial point, and $B$ is called terminal point.
It all depends on what your definition of a vector is, of course. For example, it is common to consider a vector as an equivalence class of all those oriented line segments of this kind that have the same length (magnitude) and the same direction. With such a definition, a "vector" has no initial point and no terminal point, although you can pick any initial point you like and consider the representative oriented line segment originating from that point.
In the more general setting of a smooth manifold, you often have a tangent vector in the particular tangent space (or fiber) $T_pM$ that corresponds to a point $p$ in the manifold. In such a case, you need both this foot point (is this the common name?) $p$ and a representation of the vector in the tangent space "sitting" at that point. In this general setting, there is no canonical equivalence between a vector in $T_pM$ and a vector in another fiber $T_qM$ (here $q\ne p$ is another point in the manifold $M$).
Usually, when $v\in T_pM$, we just say that $v$ is a tangent vector "at" $p$. As I said, I think I heard $p$ being called the foot point of $v$.
A: A good question. Since an important property of a vector is its direction it is hard to talk about vectors without having words for where they start and end.  
In my experience we have generally called the source or beginning of a vector its "tail" and the destination or end of the vector its "head".

A: Uh, you aren't actually talking about a vector here.  A vector is not defined by a beginning and end point.  It has a direction and magnitude (and equivalents in other coordinate systems) but no fixed origin.
The vector might have a location where the coordinates are taken.  This is particularly relevant in curvilinear coordinate systems.  If you draw a vector there, you typically draw the respective arrow with its base on the location of the vector, its shaft being straight and tangential to the coordinate directions at the origin of the vector, and the component sizes being in proportion.
Sometimes arrows are drawn for directed line segments.  Those do have a starting and an ending point.  In that case, the shafts of those arrows will follow any curvilinear coordinate system.  Visualizing a directed line segment specified by starting point and displacement vector is rarely done when more than one coordinate is actually changing since then the path along which the displacement is achieved has no obvious candidate: there is no obviously preferred route between starting and ending point any more.
