Understanding a Matrix: What does it represent? What I mean is that sometimes matrices represent linear equations, and other times vertices of polygons, etc. This gets very confusing. How would I know if someone doesn't explicitly state it, what is being represented?
 A: A matrix is simply an array.
You will know a matrix when you see one as they look like this:
$$\begin{bmatrix}a & b & c & d \\ e & f & g & h \\ i & j & k & l\\ m & n & o & p \end{bmatrix}$$
As for what it represents; that will be given in the context of the mathematics. 
But here are some general things to look out for:
For example, if you are just given on it's own:
$$ \left[
    \begin{array}{ccc|c}
      3&1&2&3\\
      7&4&5&6\\
      1&8&4&2
    \end{array}
\right]$$
then this is known as the Augmented matrix and usually means
$$ \left[
    \begin{array}{ccc|c}
      3&1&2&3\\
      7&4&5&6\\
      1&8&4&2
    \end{array}
\right]=\begin{cases}
3x+y+2z=3 \\ 
7x+4y+5z=6 \\ 
x+8y+4z=2
\end{cases}$$
Also note that if you see a matrix ${A}$ written as $$\det{A}=\begin{vmatrix}a & b & c & d \\ e & f & g & h \\ i & j & k & l\\ m & n & o & p \end{vmatrix}$$ Then you are dealing with the determinant of matrix ${A}$.
Also, if you see
$$
{B}=\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}
$$
this can be represented as three column-vectors
$$
\vec v_1=\begin{pmatrix}1\\4\\7\end{pmatrix},\vec v_2=\begin{pmatrix}2\\5\\8\end{pmatrix}\text{ and } \vec v_3=\begin{pmatrix}3\\6\\9\end{pmatrix},
$$
A: At the bare-bones-level an $(m\times n)$-matrix $A$ with real entries is a function
$$A:\quad [m]\times[n]\to{\mathbb R},\qquad (i,k)\mapsto a_{ik}\ .$$
This means that $A$ is in the first place a data structure: a family of $m\cdot n$ real numbers organized and addressed in a particular way.
It just so happens that such structures turn up every day as well in real life (spread-sheets) as in the most diverse areas of mathematics. It is generally agreed that the "visual expression" of $A$ is a rectangular array of the numbers $a_{ik}$, whereby the the numbers $a_{i1}$, $a_{i2}$, $\ldots$, $a_{in}$ constitute the $i^{\rm th}$ horizontal row of this array.
For special purposes, e.g., in linear algebra, special conventions are in force. For example, if the matrices $A$ and $B$ are both of the same "bi-size" $(m\times n)$ their sum $A+B$ is defined, and under other favorable circumstances the product $A\cdot B$ is defined. These definitions were not arbitrarily chosen by the mathematical community (note that $A+B$ is "element-wise", but $A\cdot B$ is not), but have a a well understandable interpretation in the environment were these matrices $A$ and $B$ are coming to existence.
Maybe you knew all this; but there is not more to it.
