Regular value: intuition about surjectivity condition

Let $f:M\rightarrow N$ be a smooth function between two smooth manifolds.

A $\textit{regular point}$ is a point $p\in M$ for which the differential $df_p$ is surjective.

What does the surjectivity condition for the differential mean intuitively? What is then so "regular" about this point?

What is regular about a regular point $p$ is that the fibre $f^{-1}(f(p))$ is itself a manifold in a suitable neighbourhood of $p$, and that manifold has dimension $dim (M)-dim (N)$.

About the simplest example is obtained by taking $M=\mathbb R^2, N=\mathbb R$ and $f(p)=f(x,y)=y^2-x^3$.
We have $df_{(a,b)}(u,v)=-3a^2u+2bv$ for $p=(a,b)$ and this linear form is surjective (=non-zero) unless $p=(a,b)=0$.
As a consequence the fibers of $f$, aka the contour curves $y^2-x^3=c$ are smooth submanifolds (of dimension $1$) of $M=\mathbb R^2$ for $c\neq0$.
However the contour curve through $p=(0,0) $ is the subset $C\subset \mathbb R^2$ given by $y^2-x^3=0$, which is not a manifold in any neighbourhood of $(0,0)$.

Differential geometers generally recoil in horror before beasts like $C$, whereas algebraic geometers study them under the name of singular varieties.