The Riemann sphere lets you deal with dividing by 0 by adding one more point to the complex plane: ∞[1]. Imagine putting it in the air above the origin, 0, and folding the midpoints of the four sides of your graph paper to meet there. (Yup, put all 4 points of the arrows in the same spot!) That's the Reimann sphere[2].
Going from a point the sphere back to the plane is a little weird.
1. Put a cow (point) on the top of the biodome (sphere) at infinity.
2. Pick the point on the sphere that you want to give a home on the complex plane.
3. Have the cow fire a laser beam (draw a line from infinity) through the point.
4. Follow that line back to wherever it hits the plane. That's the equivalent point on your complex plane.
Play with this a bit. Points near the top of the sphere (near the cow, at infinity) will shoot laser beams way off into the distance. Points near the bottom of the sphere will burn the grass right nearby on the plane itself.
You can then reverse the process ("If my cow were to shoot a laser at this grass, what part of the the biodome will get hit?"), to go from the plane to the sphere, because "folding up the sides of paper" doesn't accurately model what happened to make the sphere.
[1] Math with ∞ is what you'd expect: 3/0 = ∞. 3/∞ = 0.
[2] Since there is an infinite number of points on a sphere, this is entirely possible, and only mildly unsettling.
On classic, Euclid, plane a pair of parallel lines would never intersect. Given a line and a point outside of the line, there can be only one line through the point which would be parallel to the first line.
If one modify the axiom to allow for many such lines through the point (the lines which would never intersect the first line) - that would result in hyperbolic plane geometry.
If one modify the axiom to state that no such line through the point is possible (i.e. any line through the point would intersect the original line) - that would result in the Riemann sphere geometry.