In Foundations of Differentiable Manifolds and Lie Groups, by , Frank W. Warner we are given the following definitions:
Then, in Lemma 1.16 we are told: $M_m$ is naturally isomorphic with $(F_{m}/F_{m}{}^2)^{\ast}$. The symbol ${}^\ast$ denotes dual vector space.)
$M_m$ is the tangent space of to the manifold $M$ at the point $m$.
To begin with, I'm not sure what it means for $f$ and $g$ to "agree on some neighborhood". Dose that mean "in the limit as the radius of the neighborhood goes to zero", or does it mean they are exactly identical, as in the implicit function theorem?
I have some vague idea of what an ideal is. For rings, an ideal is a non-empty subset $\mathfrak{a}$ of a ring $\mathfrak{R}$ such that for $a,b\in\mathfrak{a}$ and $r\in\mathfrak{R}$, we have $a-b\in\mathfrak{a}$ and $ra\in\mathfrak{a}$.
But main question is: what does $F_{m}/F_{m}{}^2$ mean?
The basic idea of this definition is that you want to capture local variations of functions that give you the vectors. In some coordinate system where the point $m$ is 0 a generic function vanishing at $m=0$ looks like $$ f(x)=a_\mu x^\mu +b_{\mu\nu}x^\mu x^\nu+\dots $$ We mod out by $F_m^2$ precisely to remove higher order terms as $b_{\mu\nu}x^\mu x^\nu+\dots \in F_m^2$ we have $$ f(x)=a_\mu x^\mu \,\,\text{mod} \, F_m^2 $$ and we see the natural identification of the dual of $F_m/F_m^2$ with the tangent space. Note this also explains why we took only the functions vanishing at 0.
The precise definition of the quotient of a ring $R$ by an ideal $I$ is: an element $[r]\in R/I$ is the equivalence class of $r$ under the equivalence relation generated by $r\sim r+i$ for any $i\in I$. The properties of an ideal you mentioned above are precisely what you need for $R/I$ to also have a ring structure.