I just started to study about manifolds and there is something I miss about how the use of this math in a real scenario. I think the following problem will clarify my doubts and I hope your answer won't be too difficult. Imagine you are an ant (that lives in 2 dimensions) on a very big sphere and for some reason you want to describe the sphere with 2D cartesian coordinates. For every point there is a tangent plane and for this reason you can map every infinitesimal region of the sphere on some plane. I'm ok with the concept: it's like how humans use many geographic maps to describe the earth.
What I don't understand is how the ant can translate everything in formulas. For example if the ant want to describe a curve along the sphere how would it do? Obviously in this example the only things that the ant can use are the instrument to measure points on a 2D cartesian coordinate system. The ant can't use the equation of the sphere in 3D space.
The ant has to use an "atlas" to translate between names of places on the sphere ("my home", "the bank", "four ant-hills over and three up", etc.) and cartesian coordinates. Also, his "atlas" only applies to his neighborhood. If he wants to describe points in a far away neighborhood, he has to find another atlas, and has to look at the transition function from his atlas to the other neighborhood's atlas.
The last paragraph sounds silly but it is literally how a manifold is defined. The only equation the ant can use to describe the sphere is the "equation" $$ \text{sphere}=\text{the set of all points}. $$ It is like asking for an equation describing all of $\mathbb R^3$, as a subset of $\mathbb R^3$.