How are polar and rectangular planes related?

Polar coordinates (r,ϑ) and Cartesian coordinates (x,y) are related as follows:

Cartesian -> Polar
x = r cos(ϑ)
y = r sin(ϑ)

Polar -> Cartesian
r^2 = x^2 + y^2
ϑ = arctan( y/x)

The polar coo's can be extended to cylindrical by introduction a dimension z - which is perpendicular to the polar plane, it follows that:

z = z 
in between the systems (r,ϑ,z) and (x,y,z)

The polar coordinates are particularly useful when a problem has circular or cylindrical symmetry, i.e. it is usually "easier" to perform calculations on a circle (for example) using the polar coordinates.