Rn notation is a way of describing mathematical systems of numbers, and is really very simple.
The most familiar application of the Rn system is almost always given another name, such as the Cartesian Plane, two-space, or even simply a grid! That system is R2 (pronounced "R2" not "R squared"), the two representing the number of variables required to fix any one point definitely within the system. Thus in R2 the point (1,6) can only be in one spot.
However, the Rn system can extend to infinity, and comes into wider use the higher the value of n. Hence, three space is commonly referred to as R3, and once you get beyond three variables, there really is no other way of describing the system. (So a point in R4 would be (x,y,z, and another identifier)).
This system is based from a real line (hence R); it is simply a series of products of the line with itself.
As this system is usually only used in applications of mathematics, the only times you would see it would be in applications of this system (physics, pure mathematics, and sometimes business).