Ohm's law says that V=IR. This means that if voltage in the resistor increases then so does the current. Since neither term is raised to any power it means the relationship is linear. If the relationship is linear then if one plots voltage vs. current, the data should form a straight line. Thus, if the data for a device follows a straight line, the device obeys Ohm's Law and is said to be an ohmic device. Not all devices are ohmic. (Semiconductors are an example of a non-ohmic device.)
We are plotting V on the y-axis and I on the x-axis. When a straight line is drawn through the data, the slope of the line (rise over run) will have the units of voltage/current. From Ohm's Law this give units of resistance (V/I=R). Thus when we measure the slope of this data we are in fact finding out what the resistance of the little resistor is.
The larger resistor is called a rheostat and its resistance adjustable. This device is used only to vary the amount of current that comes from the power supply. The two resistors are in series, so whatever current is in the larger resistor is also in the smaller resistor. Thus if the amount of current in the reisistors is varied, the amount of voltage will also be varied. We was actually measured was the current in both resistors (using an ammeter), and the voltage across the small resistor.