My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry.

Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq X$ a nonsingular closed subvariety. Let $\mathcal{I}$ be the sheaf of ideals defined by the closed embedding $i: Y \hookrightarrow X$, and consider the sheaf $\mathcal{C}_{Y/X}=: \mathcal{I}/\mathcal{I}^{2}$. We define the normal sheaf of $Y$ in $X$ to be $\mathcal{N}_{Y/X}=:\mathsf{Hom}(\mathcal{C}_{Y/X},\mathcal{O}_{Y})$.

This definition can be found on Hartshorne's "Algebraic Geometry". Recently I came across some concrete examples of normal bundles that I cannot understand.

$\mathbf{(1)}$ Let $C$ be a nonsingular curve of degree $2$ and genus $0$ in $\mathbb{P}^{3}_{k}$. with $k$ an algebraically closed field. It can be proven that for any such curve, there exists a quadric $Q$ in $\mathbb{P}^{3}_{k}$ such that $C$ lies on $Q$. Then $\mathcal{N}_{C/Q}=\mathcal{O}_{C}(1)$, and $\mathcal{N}_{S}|_{C}=\mathcal{O}_{C}(2)$.

$\mathbf{(2)}$ Let $X$ a nonsingular irreducible cubic surface in $\mathbb{P}^{3}_{k}$, with $k$ an algebraically closed field. Let $H$ the hyperplane section of $X$. It can be proven that there exists a nonsingualr irreducible curve $C\in |4H+2L|$ of degree $14$ and genus $24$, for a line $L$ on $X$. Then $\mathcal{N}_{C/X}=\mathcal{O}_{C}(C)$, and $\mathcal{N}_{X}=\mathcal{O}_{X}(3)$.

The geometrical meaning of the twisting sheaf, and the Picard group construction are very clear to me. I don't understand how can normal sheaves be computed in the examples presented above. However, rather than focusing on those two cases, I would like to understand the nature of the normal sheaf - i.e. what does it say about the varieties? Is there a more agile definition? How do normal sheaves relate to twisting sheaves? How can they be computed? I am mosty interested in the case of regular projecive schemes of low dimension, over an algebraically closed field. Simple examples are also welcome.