'g' tells us how much force acts on a unit mass. The force we
recognize as the 'weight' of a body - F=mg .
The units for g are then (as pointed out by others) N/kg and g is a
measure of the strength of the gravitational field at that point. I
would feel comfortable calling g the 'gravitational field strength'.
This way it need not be confused with the acceleration of falling
bodies. Also the acceleration of falling bodies is affected by air
resistance and any other forces that may act on them while 'g' is
independant of all these 'details'.