1. I'm trying to understand what's a coordinate patch on a surface, I prefer a visual explanation, no need to be rigorous.

2. If I understand you correctly, you want to know what a chart is.

This has two parts:

The first part, is an open subset of the surface. Usually, we are interested in a connected region. For example, if the surface was a sphere, that represented the surface of the Earth, one such possible region might be "the northern hemisphere" (up to, but not including, the equator).

The second part, is a homeomorphism between the open set (our "region" of the surface), and an open subset of the Euclidean plane (typically containing the origin, although this is not strictly necessary). By homeomorphism, we mean a bijection that is continuous both ways.

For example, one such homeomorphism between the upper hemisphere of a sphere and an open region of the plane is projection, where we have a plane centered below the "north pole" and just ignore the $z$ coordinate.

This lets us use a "local coordinate system" on the sphere's surface, based on the "normal coordinates" of the plane.

Typically, we want to ensure different "coordinate patches" are COMPATIBLE. So, on the region where they overlap, we need to have a "transition function" that allows us to express the coordinates of one patch in terms of the other, and vice versa.

There is a nice picture of this here:

Put another way: as long as our "map" of the sphere is not "too big", we only need 2 coordinates to locate a point on the sphere within the region mapped. Similar considerations apply for other surfaces, such as the surface of a torus (doughnut), or an ellipsoid (egg). We might have a lot of "coordinate patches" if the regions we are using to describe by these are "small". We then collect all of them together in an ATLAS, which can be thought of as a "book" of "coordinate patches" (local maps of our surface).

The terminology for this is often taken from cartography, because the surface of a sphere (our planet) was one of the first surfaces we did this for, and then we have a "literal" set of charts which form an atlas (these were often used for navigation, and legal purposes, such as property determination, both on the individual, and national level).

A somewhat naive and non-technical way of saying this is: "smooth surfaces are locally flat", so we can use $(x,y)$ coordinates for a suitably limited patch of a surface (which on the actual surface may be "curvy" rather than "rectilinear").

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