|Polygonal, Subdivision, NURBS spheres from left to right|
Polygons are surfaces made up of faces, each face is an enclosure of edges, whether it be three edges, four edges, or more(n-gons). These surfaces can be made up of many faces, as many as your computer can handle or just a single face. Each edge runs in a straight line and the point where these lines meet is called a vertex. And since polygon edges flow in straight lines, they easily create hard, faceted surfaces but of course more faces are required to form smooth, curvy surfaces. Poly objects can be combined and welded together, cut, sculpted, extruded, extracted, UV mapped etc.
NURBS geometry is made of patches generated from curves. They easily form curvy rounded surfaces, placing points closer together enables the user to create sharper corners. These surfaces can be stitched together, trimmed, sculpted etc.
Subdivisional surfaces allow the user to focus on detailing regions instead of making the entire mesh heavier, hence the name subdivision. It feels like a hybrid of NURBS and polygons, easily generating smooth and rounded surfaces like NURBS while utilizing many polygonal tools.
So they are all capable of taking you to the same destination just using a different road, it's not uncommon for a modeler to convert back and forth between different geometry types. But at the end of the day it's all about your preference so try them all.
NOTE: As of Maya 2014 the subdiv surfaces tools have been removed from the Maya user interface, they are still available just hidden since the modeling method largely fell out of use.