The primary reason that manhole covers are round is because they cannot fall through the opening. Other reasons for their shape include ease of handling and manufacturing.
Circles have no angles, which means that no matter how a circle is turned, it fits flush with an opening. The diameter of a circle also insures that it has a wide middle that prevents it from fitting through a manhole opening if tipped. Shapes with angels must have their sides perfectly aligned in order to fit securely into the opening. Plus, if turned diagonally and held upright, they could be dropped into a square or rectangular-shaped manhole.
It is also easier for workers to slide round manhole covers back into place because the circular shape does not have to be positioned in any specific fashion. Should the manhole cover slip while it is being put back into place, its round shape prevents it from falling into the hole by stopping it at its widest point. It is also purportedly easier for manufacturers to produce round shapes. This is likely because manufacturing a circle does not involve angle and side calculations. Interestingly, the question about the shape of manhole covers is one that Google asks of prospective employees.