This question is related to question 2 of section 1.1 in Hatcher's Vector Bundles and K-Theory, which asks to define a quotient vector bundle. The definition can be found here: Constructing the Quotient vector Bundle. However, I am having trouble coming up with examples of quotient vector bundles.
What are some interesting and/or useful quotient vector bundles?
As written in the comments, if $Y \subset X$ is a subvariety we have an exact sequence $ 0 \to TY \to TX_{|Y} \to N_{Y/X}$.
This is very useful in intersection theory. For example, if $Y \subset X$ is a complex curve in a complex surface, you can compute $Y^2$ using the normal bundle. Indeed, this number is by definition the cardinal of $Y \cap Y_{moved}$ where $Y_{moved}$ is a copy of $Y$ intersecting $Y$ transversally. How to move $Y$ : simply move it along a section of the normal bundle. The intersection point will be exactly zeroes of this sections.
For example, if $Y = L$ is a complex line (i.e topologically a sphere) and $X$ a plane (i.e topologically $\mathbb CP^2$) we get the sequence $0 \to O(2) \to {T \mathbb CP^2}_{|L} \to N_{L/\mathbb CP^2} \to 0$ which gives (taking Chern class) $(1+2p)(1+ap) = 1 + 3p$, where $a = L^2$. So we deduce that $L^2 = 1$. Of course this is an easy consequence of Bezout theorem but for more complicate example this is a useful method.