If I understand correctly, you're asking what is (called the Riemann sphere). As a set it's simply the set of complex numbers together with an additional point you call "infinity" and denote by . As a topological space, it's the one-point compactification of the complex plane: a set is open in iff it's open in or is the complement of a compact set (in ).
This definition should be enough for you to answer the two other questions.
Intuitively, imagine you grab the real line and wrap it around a circle. It's "natural" to say the line won't cover the whole circle: there will be one point missing, so you call this point infinity. Same goes for the complex plane folded around a sphere: one point of the sphere is missing, so you just cover the hole with an extra "point at infinity". That's why it's called the Riemann sphere.
Hope it helped