I have read the following definition of cotangent bundle:
Let $X$ be a $n$-dimensional smooth algebraic variety. For any $p\in X$ there exist a neighbourhood $U_{p}\subseteq X$ and functions (named local parameters) $u_{1},\ldots,u_{n}\in \mathcal{O}_{X}(U_{p})$ such that $$ \mathfrak{M}_{q}=\langle u_{1}-u_{1}(q),\ldots,u_{n}-u_{n}(q)\rangle $$ for every $q\in U_{p}$, where $\mathfrak{M}_{q}$ is the maximal ideal of the local ring $\mathcal{O}_{X,q}$. Associating to each open subset $U_{p}$ the module $\Omega_{S/R}$ of relative differential forms of $S=\langle u_{1},\ldots,u_{n}\rangle$ over $R=\mathcal{O}_{X}(U_{p})$, gives rise to a locally free coherent sheaf. We define the cotangent bundle to be the associated vector bundle $\Omega_{X}$.
Now, suppose we have a closed immersion $Y\subset X$. We can find an open cover $X=\bigcup_{i\in I}U_{i}$ such that on each $U_{i}$, the closed subscheme $Y\cap U_{i}$ can be described as the zero locus of $r:=\mathrm{codim}(Y,X)$ regular functions $f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(U_{i})$. The conormal bundle $\mathcal{N}^{*}_{Y/X}$ of $Y$ in $X$ is defined to be the kernel of the surjective morphism $$ \Omega_{X}|_{Y}\rightarrow \Omega_{Y} $$ induced by the restriction of differentials.
My problem is that I don't understand this morphism and therefore I don't understand the definition of conormal bundle. What does "restriction of differentials" mean?
Any help (or correction in case I was misunderstanding something) would be appreciated.