What is a local normal form? I am studying manifolds from Tu's text on the subject, and he uses the term normal form but I do not recall seeing this term formally defined. The closest I've seen to a definition is Tu's statement of the constant rank theorem:

Let $N$ and $M$ be manifolds of dimension $n$ and $m$ respectively. Suppose $f: N \rightarrow M$ has a constant rank $k$ in a neighborhood of a point $p$ in $N$. Then there are charts $(U, \phi)$ centered at $p$ in $N$ and $(V, \psi)$ centered at $f(p)$ in $M$ such that for $(r^1, \ldots, r^n)$ in $\phi(U)$, $$(\psi \circ f \circ \phi^{-1})(r^1, \ldots, r^n) = (r^1, \ldots, r^k, 0, \ldots, 0). \tag{1}$$

When talking about local normal forms he references (1), so I am guessing that this is the formal definition. Likewise the Wiki on submersions has a section on local normal forms but does not formally define it.
What is the local normal form exactly and why do we care about it?
 A: As a general principle, the "normal form" of an object means something along the lines of the following: Let $X$ be a collection of objects and $\sim$ be an equivalence relation on $X$. A normal form for $x\in X$ with respect to $\sim$ is a $y\in X$ with $y\sim x$ and $y$ is the "simplest" such element in the equivalence class of $x$.
What "simplest" means is highly context dependent, often it means that using the $y$ makes calculations and arguments easiest to proceed.
In your context (one could say) $X$ is the collection of all $C^r$ maps $M\to N$ with locally constant rank at $p\in M$, and the equivalence relation is local $C^r$ conjugacy: for $f:M_f\to N_f$, and $g:M_g\to N_g$,  $f\sim g$ iff there are open neighborhoods $p\in U_f\subseteq M_f$, $f(p)\in V_f\subseteq N_f$, $U_g\subseteq M_g$ and $V_g\subseteq N_g$ and $C^r$ diffeomorphisms $\alpha:U_f\to V_f$ and $\beta: U_g\to V_g$ such that
$$\beta\circ \left.f\right|_{U_f} = \left.g\right|_{U_g}\circ \alpha.$$
In this case a projection operator is deemed to be the simplest in the equivalence class.
One practical consequence of this is that even though one has no information regarding the derivative of $f$, in appropriate coordinates once it has locally constant rank for local matters one can without loss of generality treat it as a projection operator in Euclidean space.
A: It’s not a technically defined term (well I guess one could define it technically, but there’s not much point to it). It’s just one of those things which people start saying once they know what they’re talking about. But let me try to illustrate the meaning:

*

*“Local” - hopefully this is clear. Local means “nearby” in a topological sense, i.e for every point there is an open neighborhood such that … In this case of submersions, you’re saying for every point there exist local charts such that your map has such and such properties.

*“Normal form” - this just means a “ has a typical format of” or “resembles the standard type…”. It’s just a manner of speaking in math. For example, consider the map $T:\Bbb{R}^n\to\Bbb{R}^m$, $T(r^1,\dots, r^n)=(r^1,\dots, r^k,0,\dots 0)$. We often refer to this linear map $T$ as the standard rank $r$ map. Why? Because it is clearly a linear map which has rank $r$, and it is very easy to write down. In your situation, they’re saying that if you have a submersion $f$, then after composing with diffeomorphisms on the domain and target, you can make that triple composition be equal to (a restriction of) $T$. i.e after appropriate compositions, every submersion is equal to $T$.

In linear algebra you’ll see many more examples. For example, you may have heard of Jordan canonical/normal form or rational canonical form. What do these terms mean? It means under appropriate hypotheses, given a linear map $T:V\to W$, there exist bases $\beta,\gamma$ on the domain and target respectively such that the matrix representation $[T]_{\beta}^{\gamma}$ has such and such simple entries (e.g. some number of Jordan blocks).
Another example: in symplectic geometry, the things of study are $(M,\omega)$, where $M$ is a smooth manifold, and $\omega$ is a $2$-form on $M$ satisfying certain conditions (closed, and non-degenerate), known as the symplectic form. The standard and simplest example is $(\Bbb{R}^{2n},\omega_0)$, where we label the Cartesian coordinates as $(q^1,\dots, q^n,p_1,\dots, p_n)$, and define
\begin{align}
\omega_0=\pm\sum_{i=1}^ndq^i\wedge dp_i;
\end{align}
sign conventions are a nightmare so just ignore that. Now, it is a wonderful theorem of Darboux that for any symplectic manifold $(M,\omega)$, and for any $x\in M$, there is a chart $(U,\phi=(q^1,\dots, q^n,p_1,\dots, p_n))$ around the point $x$ such that on $U$, we have $\omega=\pm\sum_{i=1}^ndq^i\wedge dp_i$. Said diffferently, $\omega=\phi^*\omega_0$. So, around every point, there’s a chart such that the chart makes the abstract symplectic form $\omega$ look like the plain old vanilla symplectic form $\omega_0$.
So you see the recurring idea: you start with some simple object which you understand very well (e.g. a linear map of rank $r$, or a Jordan block, or a diagonal matrix, or the identity map or whatever). Then you take a look at your new complicated object (could be a random linear map, or a submersion, or a local diffeomorphism, a tensor field, a differential form etc etc), and then you say ok, this object I have is complicated, but after some finessing, I can make it look like the simpler object. You can certainly try to formalize this notion using equivalence classes and stuff, but hopefully the idea is clear.
