How are matrices related to vectors? I know it's a silly question, but there's something I feel like I'm missing in my understanding of matrices. 
I'm studying linear algebra, and much of what we covered in the first few topics related to vectors (vector spaces, linear independence, etc.), but then all of a sudden we started using matrices, loosely defining them as an "array of numbers". 
So I'm kind of confused, is a matrix supposed to be a list of vectors? And if so, are they the rows or the columns of the matrix? 
 A: If 

all of a sudden we started using matrices

then your instructor may not have provided enough motivation.
You seem to know something about vector spaces - abstractly, they are spaces of things you can add together and multiply by scalars. Since you can do that with matrices (of the same shape), the set of matrices is itself a vector space.
But it's much more than that. In linear algebra a matrix can be seen as a way to describe a function from vectors in one vector space to vectors in another. Since you can compose functions (you did that in calculus) you can apply one matrix and then another. That leads to the study of how to multiply matrices (something you can't do with vectors).
Sometimes matrices enter the linear algebra course as a way to write the coefficients of a set of linear equations without having to write the variable names. Then they are (for a while) just arrays of numbers.
This should all become clearer to you as the course progresses. 
A: Matrix is made up of vectors which are both rows and columns of matrix.studying matrices using vectors help to understand the various properties of matrices.vectors are what makes linear algebra applicable to matrices.
As far as definition is considered matrix can be said   as both a collection of vectors or a array of numbers.Both are different ways to see same thing
A: A vector is a linear array of quantities.
A matrix is a 2-dimensional array of quantities.
Three dimensional and higher dimensional arrays also exist, they are called Tensors.
A matrix can be thought of a sequence of column vectors, but also as a sequence of row vectors, both interpretations are useful.
 An example of an important theorem in this regard is :row rank=column rank.
A: Have you started multiplying matrices and vectors yet? If so, one can look at it from a more abstract point of view. Informally, if we take the set of all vectors with $n$ entries, we can look at the kind of mappings from that set to itself. That is, we want to find a function that takes a vector with $n$ entries ($\mathbb{R}^n$ if you are working with reals) as input, and outputs another one with $n$ entries$^1$. This is a large set of functions, so we might want to restrict to a certain class of functions. In particular, if we limit ourselves to functions that are linear, it turns out that the only remaining functions are exactly those that correspond to multiplying a vector with an $n\times n$ matrix! This gives us a nice relation between vectors and matrices, matrices map vectors linearly to other vectors, and multiplication of matrices then corresponds to composition of linear mappings.
$^1$More general, something similar also holds for mappings between any $\mathbb{R}^n$ and $\mathbb{R}^m$.  
A: Fundamentally, vectors and matrices are different things.
A vector, e.g., $\mathbf{v} \in \mathbb{R}^n$, is a numerical entity in an $n$-dimensional space. A matrix, e.g., $\mathbf{A} \in \mathbb{R}^{m \times n}$, is a linear transformation from a $n$-dimensional to a $m$-dimensional space. In other words, if $T\left\{\cdot\right\}$ is a linear transformation, then there exists a matrix $\mathbf{A}$ such that $T\left\{\mathbf{v}\right\} = \mathbf{Av} = \mathbf{w} \in \mathbb{R}^m$.

*

*So why so many people states that "vector is just an one-column matrix"?

Some points:

*

*Many people, actually, don't even have the insight that $\mathbf{Av}$ is a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$.

*Denoting a vector as a column is a mere convention, indeed. Consequently, they are prone the think a vector as a particular case of a matrix when there is only one column.

*My definition holds only for linear algebra analysis. For multilinear algebra, a matrix (or a tensor) is a numerical entity as well as a vector.

So, in nutshell, vectors are numbers, and matrices are how these numbers are (linearly) transformed. A good reference the book of the renowned Professor Gilber Strang. In section 7.2, he discusses that a matrix is actually a linear transformation. I seize this difference only when I could see (literally) how matrix transforms spaces, on the Youtube videos of 3blue1brown, that is utterly beautiful.
I am not saying that seeing a vector merely as a one-column matrix is wrong though. But, IMHO, it is a shallow interpretation.
A: A matrix has several interpretations. The first is as an array of numbers, which we can turn into a vector in the usual way by defining addition component-wise and scalar multiplication in the usual way. It's the simplest interpretation but this turns out to be rather powerful.
The second interpretation of a matrix is as the representation of a linear transformation $T: U \rightarrow V$ with respect to some basis, lets call it $M$ and this is where matrix multiplication comes into play. So say I have another transformation $S: V \rightarrow W$ represented by $N$ then the composition $S(T(u))$ is represented by the matrix $NM$. When all the maps considered are endomoprhisms the matrices will be square and that gives an even richer algebraic structure of its own.
Finally, we can consider a matrix as a multilinear object. Say we choose a column vector and only allow linear operations on that column but no others. The whole matrix isn't linear but that one column is. Now extend that idea to every column so that each column is a linear space in isolation but it's not linear as a whole. I could have also done this with the row. You may be familiar with such an interpretation if you've come across the determinant, which is alternating multilinear form.
Tensor products abstract this notion so that the columns or rows no longer have to be the same size so you can put a $2\times 1$ matrix next to a $3\times 1$ matrix and work in this manner. I think that a matrix gives a simple example of what a tensor product can look like.
