The drain characteristics of a MOSFET are drawn between the drain current ID and the drain source voltage VDS. The characteristic curve is as shown below for different values of inputs.
Actually when VDS is increased, the drain current ID should increase, but due to the applied VGS, the drain current is controlled at certain level. Hence the gate current controls the output drain current.
Transfer characteristics define the change in the value of VDS with the change in ID and VGS in both depletion and enhancement modes. The below transfer characteristic curve is drawn for drain current versus gate to source voltage.