The points highlighted in this image (produced by @AnindoGhosh):
Are called Fiducial markers.
A fiducial marker or fiducial is an object placed in the field of view of an imaging system which appears in the image produced, for use as a point of reference or a measure. It may be either something placed into or on the imaging subject, or a mark or set of marks in the reticle of an optical instrument.
Basically, most modern electronics are assembled by a robotic system. To place the electronic components in exactly the correct place on a circuit board, the robot needs to know exactly where the circuit-board is with reference to it's robotic arm.
These markers are designed so that they're easy to measure using a computer-vision system. Basically, the pick-and-place robot has a camera on it's arm, and it takes a picture of each fiducial, performs a centroid operation on the little circle in the middle, and uses the centroid's results to establish the precise position of each dot. Since there are two dots, it can derive a precise position (both X-Y and R (rotational)) in a 2-dimensional plane.
The reason the soldermask (the blue covering over most of the PCB) is held back from covering the fiducial is that the components need to be correctly aligned to the copper layer (it's composes the actual pads), not the solder mask. Generally, there is some mis-match between the soldermask, the copper, and the silkscreen(the white printing) called "registration error".
Article on fiducials from LadyAda.
