My question is with regards to understanding blowing up a variety, $X$, along an arbitrary subvariety, $Y$, with respect to the conormal bundle. I have been told that a blow up in essence is replacing the subvariety $Y$ by the projectivized conormal space for each point in $Y$. My understanding of the conormal space is to be the dual of the normal space at that point, which is defined by $N_p = T_p(X)/ T_p(Y)$, with $T_p(X)$ the affine tangent space. So, the fibers $\pi^{-1}(p)$ for a point $p \in Y$ will be isomorphic to $(N_p)^*$.
Please let me know if my understanding of this is even correct and perhaps explain this through simple examples of blow ups like blowing up a point in projective space, or blowing up the cuspidal cubic or the nodal cubic as I am most familiar with these examples.
I noticed on many posts here that Eisenbud and Harris' Geometry of Schemes book is highly recommended for understanding blow ups, but I would prefer to understand this without the language of schemes or sheaves (just language of quasiprojective varieties and vector bundles). Thank you for any help you are able to provide!
EDIT: I am currently learning from Shafarevich's Basic Algebraic Geometry textbook (3rd ed) which deals with blowing up at a point in projective space or a quasiprojective variety (Chapter 2 Section 4) i.e. $\sigma: \Pi \to \mathbb{P}^n$ where $\Pi \subset \mathbb{P}^n \times \mathbb{P}^{n-1} = \{ ([x_0: \cdots : x_n] , [y_1 : \cdots : y_n] ) \}$ defined by $x_iy_j = x_jy_i$ for $1 \leq i,j \leq n$. with center $\xi = [1 : 0 : \cdots : 0]$