Matrix Multiplication If I had any $m\times n$-matrix, where $m$ and $n$ are variables, what can I multiply this matrix by? 
Can it be multiplied by itself?
By a square matrix?
Another $m\times n$-matrix?
Or can it be multiplied by a scalar?
 A: It can be multiplied componentwise by a scalar. It can be multiplied on the right by any $n$-row matrix, and on the left by any $m$-column matrix.
A: $A\cdot B$ is defined, iff $A$ has as many columns as $B$ has rows
So, if $A$ is a $n\times m$-matrix and $B$ is a $m\times p$-matrix the product $A\cdot B$ is well-defined and the result is a $n\times p$-matrix.

Moreover you are allowd to multiply any matrix by a given scalar.
A: Just to add a little to the answers here... there are a few details missing here but this is a good reference for the details. 
If the entries of your $m\times n$ matrix $A=[a_{ij}]$ lie in a ring $R$ (you can think $\mathbb{R}$ or $\mathbb{C}$ probably) then your matrix can be seen as a linear map, $T_A$:
$$T_A:R^n\rightarrow R^m\,,\,e_j\mapsto \sum_ia_{ij}e_i,$$
where $a_{ij}$ is the $(i,j)$-entry of $A$. That is $T_A$ takes in $n$-scalars and 'spits' out $m$.
You can show that the composition of linear maps as matrices is given by the ordinary matrix multiplication.
Now you can compose (multiply) with a linear map $T_B$ (matrix $B$) on the left or the right. Consider first on the left: $BA$. Now this is a linear map given by $T_B\circ T_A$:
$$R^n\underset{T_A}{\longrightarrow} R^m\underset{T_B}{\longrightarrow} V,$$
where $V$ is some vector space. If $B$ is to be a matrix with entries from $R$ then $V=R^k$ for some $k\in\mathbb{N}$.
Therefore we have:
$$R^n\underset{T_A}{\longrightarrow} R^m\underset{T_B}{\longrightarrow} R^k,$$
and so $A$ can be multiplied on the left only by a matrix $B$ that induces a linear map
$$T_B:R^m\rightarrow R^k;$$
i.e. $B$ is a $k\times m$ matrix. Therefore to multiply $B$ by $A$ on the left we have
$$\underbrace{B}_{k\times m}\quad\cdot\quad\underbrace{A}_{m\times n};$$
i.e. what the other answers said. Note that $T_B\circ T_A:R^n\rightarrow R^k$ so that $BA$ is a $k\times n$ matrix.
You can show that for multiplication on the right you need an $n\times k$ matrix.
