What is the significance of the quotient $m/m^2$?

If ${\mathcal{O}_{X,p}}$ is the local ring of a variety/scheme/manifold at a point p and ${\mathfrak{m}_p}$ its maximal ideal, then the cotangent space of $X$ at p is ${\mathfrak{m}_p}/{\mathfrak{m}_p}^2$. We also have $\operatorname{Hom} \left( {{\mathfrak{m}_p}/{\mathfrak{m}_p}^2,k} \right) \cong \operatorname{Der} \left( {{\mathcal{O}_{X,p}},k} \right)$ where k is the residue field at p.

Consider the local ring $R = C_{{\mathbb{R}^n},0}^\infty $. Then $R/{\mathfrak{m}^2}$ splits canonically as $\mathbb{R} \oplus \mathfrak{m}/{\mathfrak{m}^2}$. So consider $f \in R$ and look at the Taylor expansion $f\left( {{x_1},...,{x_n}} \right) = f\left( 0 \right) + {\sum {\left. {\frac{{\partial f}}{{\partial {x_i}}}} \right|} _0}{x_i} + r\left( {{x_1},...,{x_n}} \right)$.

See $f\left( 0 \right)$ lies in $\mathbb{R}$, the remainder $r\left( {{x_1},...,{x_n}} \right)$ lies in ${\mathfrak{m}^2}$, and we are left with the differential $\operatorname{d} f = {\sum {\left. {\frac{{\partial f}}{{\partial {x_i}}}} \right|} _0}{dx_i}$ in our cotangent space (where the differential map is viewed as the natural projection $\operatorname{d} :R \to \mathfrak{m}/{\mathfrak{m}^2}$ under this splitting).

The quotient can be made into a vector space over $A/\mathfrak{m}$. In this case, it is the Zariski Cotangent space (after localization) and is isomorphic to the tangent space for an algebraic variety.

Lets consider curves. one can view the quotient $k[x,y]/\mathfrak m$ as a co ordinate ring (functions on a variety) around a point associated to the maximal ideal (by nullstellensatz. Elements in the maximal ideal are the interesting functions (polynomials) that survive after forming the local ring. Higher degree polynomials are nonlinear, so modding out any higher degree polynomials leaves only the linear functions defined around your point. The space of linear functions through a point defined on a variety is exactly the cotangent space.

here is a slightly more involved discussion of regularity conditions in terms of the cotangent space (on a blog post I jotted down some months ago.)