English names for vector beginning and end

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".

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.

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 $v$.