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.