where to place grounds on a circuit
Ground can be just the convenient name of a circuit-node that is commonly connected by several or many components in a circuit. Nodes called ground hardly ever connect to a physical wire that is grounded (think battery powered equipment and aeroplanes). In the link for ground above, the wiki page opening paragraph states this: -


  In electrical engineering, ground or earth is the reference point in
  an electrical circuit from which voltages are measured, a common
  return path for electric current, or a direct physical connection to
  the earth.


So, it can be a direct physical connection to the earth or, more than likely, it's a reference point.


  I see circuits with multiple grounds


And consider that for many circuits a 5 volt node symbol is used for components that make a connection to the positive power rail of 5 volts. Using a symbol to replace a wire can make complex diagrams easier to read.


  what determines the number of grounds and where should they be placed?


If we are talking about the ground node that is often called "0 volts" then you can use the symbol anywhere that it makes the reading of the circuit easier.

However there several different symbols that are called ground (loosely) that may exist on the same circuit: -



Above picture taken from here and I would recommend that page for further reading.

Then there is protective earth and that has a specific meaning for safety: -



Apart from the protective earth, any of the symbols above can be used to represent a common-circuit-node and not necessarily mean a direct connection to a ground rod. Some have more of a specific meaning (like earth and chassis) but the rest don't necessarily infer an electrical bond to our planet.