What is the name for a vector with a single non-zero component? 
What do you call a vector with a single component, e.g. $[a, 0, ..., 0]$ or $[0, b, 0, ..., 0]$, where $a, b$ are any non-zero number? 

I'd like the language to differentiate from a vector with many components, e.g. $[a, b, 0, c, 0]$.
 A: I would call such vectors 'multiples of the standard basis vectors'. The standard basis vectors are normally written as $e_i=(0,0,\ldots,0,1,0,\ldots,0,0)$ where the $1$ appears in the $i$-th position, and so the vectors you are considering are equal to $ae_i=(0,0,\ldots,0,a,0,\ldots,0,0)$ for some $a$ and $i$.
A: Having a single non-zero component, as do, e.g., $[a, 0, 0, . . ., o]$, $[0, b, 0, . . ., 0]$, etc., is a basis-dependent property, so it does not really reflect the coordinate independence which makes many abstract vector methods so powerful.  It is easy to see that, under a suitable change of basis, these vectors will no longer have exactly one non-zero component.  So, I think I would adopt a somewhat prosaic description, such as vectors with a single nonzero component in the such-and-such basis; here you can think of "such-and-such basis" as a verbal variable ranging over the domain consisting of any and all bases for the vector space under consideration! ;)
The notion of standard basis vectors, as mentioned by Daniel Rust in his answer, can be a helpful one here, but in many instances it is not clear exactly what the standard basis is; for example, when we are dealing with $\Bbb F^n$ for some field $\Bbb F$, the standard is clear: $[1, 0, 0, . . . , 0]$, $[0, 1, 0, . . . , 0]$ etc.  But in some contexts, there really is no standard; I am thinking here of bases for the tangent spaces on a manifold; this is a situation is which adopting a standard is not necessarily possible or advantageous.
Hope this helps.  Cheerio,
and as always, 
Fiat Lux!!!
A: You may call it a single-entry vector:
https://en.wikipedia.org/wiki/Single-entry_vector
A: Typically in Machine Learning context they are usually called "One-hot vectors" and they are used to encode which variable gets activated each time, assuming only one can be active.
