For a local ring $R,$ how do we define this operation? Let $R$ be a commutative local ring with maximal ideal $m.$ Then $R/m = k$ is a field. And apparently my professor says that $m/m^2$ is a $k-$vector space. I can see that $m/m^2$ satisfies the abelian group properties of a vector space. But what about scalar multiplication? Consider an arbitrary scalar of $k, r + m.$ So multiplying this with an arbitrary 'vector' gives us $(r + m)(x + m^2) = rx + rm^2 + mx + mm^2.$ Clearly, $rm^2 + mx + mm^2$ is a subset of $m^2.$ However, is it equal to $m^2?$ To my knowledge, $rm^2 + mx + mm^2$ must be equal to $m^2$ for us to call the product $rx + m^2.$ Or is there another way to define scalar multiplication? 
 A: It is not necessarily equal to $\mathfrak m^2$, but it is a subset thereof, hence the congruence class of $rx$ modulo  $\mathfrak m^2$ is defined without ambiguity.
A: For quotient ring of the type $R/I$ we have elements of type $r+I$ where $r \in R$. Now this has multiplication of cosets in the following way 
$$(r_1+I)(r_2+I)=r_1r_2+I.$$
Similar way your $k$-vector space or $R/m$ vector space $m/m^2$ has multiplication of scalar as follows
$$(r+m)(m_1+m^2)=rm_1+m^2.$$
A: 
$(r + M)(x + M^2) = rx + rM^2 + Mx + MM^2.$

I'd like to reiterate that the line above is indicative of a misconception. The operations here and between cosets in general do not have anything to do with distributing the original operation "setwise" somehow.
The real definition is: $(r+M)(x+M^2):=rx+M^2$. There isn't anything in between. There isn't any distribution because the $+$ in "$r+M$" is just a part of the notation. It is reminiscent of the original $+$ from $R$, but it is no longer indicating an operation when we write $r+M$, which means a single element of the set of cosets. 
The general construction is that if you have an $R$ module $A$, then $A$ is also naturally an $R/ann(A)$ module, where $ann(A)$ is the annihilator of the module $A$.
In your case, $M$ annihilates $M/M^2$, so $M$, already being an $R$ module, now has its natural $R/M$ module structure as well. For a local ring with maximal ideal $M$, $R/M$ is a field, so that is where the vector space structure on $M/M^2$ arises from.
