What does the metric matrix G tell us here Let $\phi:U \rightarrow S \subseteq \mathbb{R}^3$ be a chart from $U \subseteq \mathbb{R}^2$ to a surface $S$.  $G = g_{ij}$ be the metric matrix such that $ g_{ij} = \frac{\partial \phi}{\partial x_i} \cdotp \frac{\partial \phi}{\partial x_j}$.
What information does $G$ give us?
Edit: I have found that if two maps from the same $U$ have the same $G$ then there exists an isometry between them. However this does not increase my geometrical understanding of $G$.
 A: Your metric matrix $G$ is also sometimes referred as the first fundamental form (for surfaces in $\mathbb{R}^{3}$), or, more generally, as metric tensor. 
This matrix is used to define inner product on all the tangent planes of $S$.
As you can image, defining inner product opens a huge number of applications for $G$. 
Knowing how to define inner product allows us to make use of the concept of orthogonality, measure angles between (tangent) vectors, to induce norm, and much, much more!
The very limited list of things you can do using $G$ includes


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*calculate length of a curve on the surface 

*calculate area of an appropriately defined region of the surface

*express directional derivates of functions defined on $S$

*$\cdots$



Let me also list a few cool properties of metric tensor:


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*if $G$ is positive-definite, then $S$ is a Riemannian Manifold  

*the determinant $\left| \det(G) \right|$ is equal to the volume of the parallelepiped spanned by the basis vectors of tangent space

*thus $\left| \det(G) \right|$ pops up when computing volumes or integrating over the volume

*metric tensor $G$ allows to compute geodesic lines between any two points on $S$.


*

*notion of space and geodesics allows to view Riemannian manifold $S$ as metric space


*metric tensor is used to define Levi-Civita connection


*

*this means that the notion of parallel transport can be introduced on the surface $S$


*metric tensor is invariant under the choice of system of coordinates

*the choice of metric tensor is not unique

*$\cdots$

