What are examples of vectors that are not usually called vectors? In algebra, a vector is an element of a vector space. An example of such an element is a matrix.
In linear algebra, a vector is a shorthand name for a $1 \times m$ or a $ n \times 1 $ matrix. (Whereas a matrix itself is also a vector, by definition, but rarely referred to as such.)
In (analytic) geometry, a (euclidean) vector is a geometric object with a magnitude and direction. These can be represented by tuples.
Both matrices and tuples are clearly vector elements in some spaces, but are for some reason not called vectors by name, unlike euclidean vectors and row vectors.
Are there any other examples of vectors that are not called vectors, like matrices and tuples?

Quaternion's coefficients: are just 4 real numbers, so just a vector in $\Bbb R^4$. Note, I explicitly told "coefficients" because in reality a quaternion is a extension of the complex numbers.
Sorry for the short answer, but it cannot be longer than that without repeating something that is already in other answers :).
