What exactly is a matrix transformation? I'm confused on the notation for $\mathbb{R}^{m}$ and $\mathbb{R}^{n}$ and its relationship to linear transformations. We say that $\mathbb{R}^{m}$ is just the space of column vectors since there are n columns, right? 
If so, then what do we mean when we say T: $\mathbb{R}^{m} \mapsto \mathbb{R}^{n}$ is equivalent to $T_A(X) = AX$? I watched a video on linear algebra (essence of Linear Algebra) that (only if I'm remembering correctly) that all matrix multiplication was is the change of the standard basis vectors such as $\hat{\imath} \space \hat{\jmath}$ landed on different spots in the same dimension. 
What is exactly going on here? Can you have linear transformations that don't depend on the number of rows or columns to be the same? Geometrically, what is going on? It seems like I have no idea what is going on in a matrix at all. 

I know it as "linear map". It is a kind of function between vector spaces. 
If $v,w$ are vectors of the vector space $(X,K)$ and $\lambda,\mu$ are elements of the field $K$ then we say that the function $f:(X,K)\to (Y,K)$ between vectors space over the same field is a linear map if
$$f(\lambda v+\mu w)=\lambda f(v)+\mu f(w)$$
In your case $\Bbb R^m$ and $\Bbb R^n$ are the vector spaces, both over the field $\Bbb R$. There are a lot of linear maps, by example the limit function defined as
$$\lim:c(\Bbb R)\to \Bbb R,\quad (x_n)\mapsto x$$
where $c(\Bbb R)$ is the vector space of all convergent sequences composed by elements of $\Bbb R$. For convergent sequences $(x_n)$ and $(y_n)$ and $\alpha,\beta\in\Bbb R$ we have that
$$\lim(\alpha x_n+\beta y_n)=\alpha\lim (x_n)+\beta\lim (y_n)$$
