Resistance of a MOSFET in saturation mode
For a given gate-source voltage and above a certain drain-source voltage, the MOSFET behaves like a constant current sink. Say you have 3 volts applied to the gate with respect to the source - if if the drain-source voltage rises to 2 volts, the drain current reaches a plateau of about 5 milliamps. If you increase the drain-source voltage (whilst keeping gate voltage at 3 volts), the drain current remains largely at 5 mA.

This is broadly referred to as "constant current" mode of operation.


  Is the current drawn determined by the transistor or the load itself?


The current is determined by the transistor mainly but, because there is a slight slope there is a small change in current with a big change in drain load resistance.


  for example, at saturation is this the max current it can draw or the
  amount of current it will draw?


This is the typical current it will draw for a given gate-source voltage (3 volts in my example).