"103" is standard resistor (and other component) numbering - the first two digits specify a value and the third is a power-of-ten multiplier:
1 0 3 = 10 * 10^3 = 10K
4 7 2 = 47 * 10^2 = 4.7K
For devices that are typically measured in thousandths, millionths or smaller (like a capacitor) then the third digit is a minus power-of-ten:
1 0 6 = 10 * 10^-6 = 1u
That black disc labelled '103' is a thermistor - a temperature-controlled resistor. It is most likely NTC, or negative-temperature coeffiecient, thus the resistance decreases as the temperature increases. This is contrary to typical physics where an increase in the temperature of a generic conductive material results in an increase in its electrical resistance.
The LM358 is included to buffer the thermistor from the Arduino/microcontroller. In simple terms, this means that the effective resistance of the analogue input pin does not have to be taken into consideration when converting the voltage measured to into a value for the temperature.
Without a better picture of both sides of the board, or better yet a circuit diagram, it is not possible to determine what the voltage on pin
S will be for a given temperature. However, it seems that the thermistor is in series with another 10K resistor (
R3, marked '103' of course) such that an increase in temperature results in an increase in voltage. At roughly 20 degrees Celsius (it depends on the manufacturer and their specifications) the resistance of the thermistor will be 10K, thus the voltage at
S will be
Vcc/2. Calculating the voltage for different temperatures requires you to know the temperature coefficient, which can be found from the datasheet or by measuring the resistance at various known temperatures and plugging those values into an equation.