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?
 A: You have mentioned matrix space as a vector space. The answer by Math_QED has given three frequent examples. One other often-mentioned example that immediately comes to mind is a field $\mathbb K$ as a vector space over a subfield $\mathbb F$.
For instance, $\mathbb R$ is a vector space over $\mathbb Q$, where each real number is a "vector" and each rational number is a "scalar". Vector addition is defined by the usual addition of real numbers, and scalar multiplication is also the usual multiplication of real numbers, but only the multiplication of a "vector" (a real number) by a "scalar" (a rational number) is considered a legitimate scalar multiplication. This vector space is infinite dimensional, as explained by a popular posting on this site.
Another very important example, which is somewhat similar to but not exactly the same as what Aloizio Macedo describes in his answer, is the concept of a free vector space.
Let $X$ be any set, and $\mathbb F$ be any field. For any $x\in X$, let $f_x:X\to\mathbb F$ denotes the membership function defined by $f_x(x)=1$ (the $1$ here is the identity element in $\mathbb F$) and $f_x(y)=0$ (the zero element in $\mathbb F$) if $y\ne x$. Then the linear span of $\{f_x: x\in X\}$, with the usual addition of functions and multiplication of a function by a scalar, is a vector space over $\mathbb F$. In other words, every "vector" here is a function $f:X\to \mathbb F$ whose preimage $f^{-1}(\mathbb F)$ is a finite set.
The moral here is that every (yes, every) set $X$ can be made a vector space over any field $\mathbb F$, even when $X$ and $\mathbb F$ are completely unrelated. Whether we can define a vector addition and a scalar multiplication that makes the vector space interesting or useful is another matter.
The concept of free vector space is useful to our understanding of the concepts of universal mapping and tensor product, but I shall not go into details here, as this is covered in many books on multilinear algebra.
A: In the "linear algebra" sense, scalars (numbers) are "vectors" of dimension 0 and all tensors are "vectors".
A: Everything.
Given any set $S$, you can consider the space of (formal) finite linear combinations of its elements $\{a_1\mathbf{x}_1 + a_2\mathbf{x}_2 + \dots + a_k\mathbf{x}_k: k\in\mathbb{N}, \mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_k \in S  \}$. This has a vector space structure.
A: The set of operators between vector spaces is itself a vector space.
The set of functions mapping vectors to the field on which they live is a vector space, as well, but this is just a special case of the first statement.
A: An example that is often hard to get for students on first encounter is given by vectors and spinors in particle physics: The representations of the Lorentz group $SO(1,3)$ fall into two classes, having integer or and half-integer spin, called vectors and spinors, respectively. Hence, you find statements such as "$\psi$ is a spinor, not a vector!" This can be particularly confusing as both are elements of a vector space, both are written as rows or columns of (commonly four) numbers (well, possibly anticommuting numbers), and both appear together in typical particle physics applications.
This is actually a general feature of the orthogonal groups, related to universal covering groups and projective representations. In particle physics, these classes of representations are used to desribe very different classes of particles, fermions vs. bosons.
A: Of course there are: A few examples that come to mind:
1) The vector space of all continuous functions with the usual function addition and scalar multiplication
2) The vector space of all sequences $u_n: \mathbb{N} \rightarrow \mathbb{R}$
3) Polynomials
The examples above are vectors in specific vector spaces, but still, we prefer to call them functions (1), sequences (2), polynomials (3).
You can think of more exotic examples, but I thought these were good examples to answer your question as these are things you have already encountered.
A: Any ordered set of numbers (or other values) can be considered a vector; it's just an abstraction.
For example, in the field of data processing, it's common to consider graphics/audio files as vectors (or sets vectors) with thousands or millions of elements each, then use the tools of Linear Algebra to manipulate them.
A: An exotic example with an interesting application: $\mod 2$ homology.
The formal sums of faces/edges/vertices of a polyhedron with coefficents in $\Bbb F_2$ are called $k$-chains $\mod 2$ ($k =$ dimension of objects) and form a vector space over $\Bbb F_2$.
See Proofs and Refutations by Imre Lakatos for a proof of the Euler's polyhedron formula using $\mod 2$ homology.
A: Anything.
Take any set $X$. Consider the set $A:=\{f:\{X\} \to \mathbb{R}\}$. $A$ has an obvious vector space structure. Now, let $\mathcal{C}:=(A-\{f_1\}) \cup \{X\}$, where $f_1$ maps $X$ to $1$. Then $\mathcal{C}$ also has an obvious vector space structure (the one which makes the "identity" an isomorphism), and $X$ is an element of $\mathcal{C}$, hence a vector.
This answer is just to emphasize that being a vector is not a quality in itself. The importance is the whole structure (the vector space, the sum, the multiplication by scalar etc) and its relevance to the situation which is being applied to, which is why we don't refer to a lot of things as vectors even though they are (because being a vector is not relevant to the situation at hand).
A: A vector space $V$ over the field $F$ can be more formally described as a set with some properties. These properties include closure under addition, and closure under multiplication with a scalar in the field $F$.
Many concepts can be thought of as vector spaces. Whether it is useful to think of them as vector spaces is another question.
Here are some examples:
Real numbers
The set of real numbers is an infinite dimensional vector space over the field of rational numbers. 
The set of real numbers is a 1 dimensional vector space over the field of real numbers.
Rational Numbers
The set of rational numbers is a 1 dimensional vector space over the field of rational numbers. It is not a vector space over the field of real numbers because it is not closed under scalar multiplication. It is not a vector space over integers because integers does not make a field.
Functions
If we take the co-domain to be the set of real numbers or the set of complex numbers:
The set of functions over a set domain is a vector space over both the real and rational numbers.
The set of bounded functions over a set domain is a vector space ...
The set of [insert description here] functions is a vector space over the real numbers/rational numbers that you can add any two such functions, or multiply one of these functions with a real/rational and still get a function that fits the description. For example: continuous, uniformly continuous, integrable, $C_n$, polynomials, rational functions, etc.
Magic squares
The set of magic squares is a vector space (I believe 4 dimensional) over the real numbers. It is also a vector space over rational numbers
Equations
The set of equations is a vector space.
Regarding fields
Ultimately, you can come up with some really silly example of vector spaces if you use some really unfamiliar/obscure fields. For example, this field. 
A: 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 :).
A: The quality of a "vector" you are describing is the ability to write it as a set of coordinates with respect to some basis of the space in which they reside.  In finite dimensions, provided that you maintain the ordering of the basis elements, this takes the form of a "tuple", or a row or column matrix. This "picture" of a vector space gets somewhat complicated once you move past finite dimensions, since our basis now contains infinitely many elements. 
Sometimes we can still write the vectors in our space as some kind of ordered list (like the space of all real sequences) even though the vector space is infinite dimensional. However we cannot write down a basis for these spaces, but we know that one must exist (assuming the axiom of choice).  For such vector spaces, the basis becomes fairly useless, and we have to adopt other tools to study them.
There are many examples of vector spaces which do not "look" like vectors in the traditional sense. The set of functions from a given set into the real or complex numbers will do, where we take the "pointwise operations" of addition and scalar multiplication.
A: A physical vector space is the one in which quantum states lives, namely a Hilbert space, which is a vector space of square integrable functions. When the space dimension is infinite, then the vectors are just functions square-integrable.
