I am trying to teach myself what the Zariski tangent space is using the following set of notes: https://math.mit.edu/~mckernan/Teaching/09-10/Spring/18.726/l_8.pdf
Let $X$ be a variety and let $p \in X$ be a point of $X$. The Zariski tangent space of $X$ at $p$, denoted $T_p X$, is equal to the dual of the quotient $$ \mathfrak{m} / \mathfrak{m}^2, $$ where $\mathfrak{m}$ is the maximal ideal of $\mathcal{O}_{X, p}$. Note that $\mathfrak{m} / \mathfrak{m}^2$ is a vector space. Suppose that we are given a morphism $$ f: X \longrightarrow Y, $$ which sends $p$ to $q$. In this case there is a ring homomorphism $$ f^*: \mathcal{O}_{Y, q} \longrightarrow \mathcal{O}_{X, p} $$ which sends the maximal ideal $\mathfrak{n}$ into the maximal ideal $\mathfrak{m}$. Thus we get an induced map $$ d f: \mathfrak{n} / \mathfrak{n}^2 \longrightarrow \mathfrak{m} / \mathfrak{m}^2 $$
At the moment, I am still unable to see why $ \mathfrak{m} / \mathfrak{m}^2, $ a vector space? What is it a vector space over?
For some background, I have not taken commutative algebra, so I am just learning whatever I need as I read these notes. I know what localization is and that $m$ is the unique maximal ideal.