Beta value of the thermistor
The classic approach when dealing with RTD's is the Steinhart-Hart equation 

\$\frac{1}{T} = a b\cdot ln(R) + c\cdot ln^2(R)\$

This is a very accurate model of the characteristic change with respect to temperature. 

This however is quite "complex" to deal with in real-time (if accurate temperature is require).

The \$ \beta model\$ is a special/reduce case of the Steinhart-Hart equation 

\$ R = R_0 e^{\beta ( \frac{1}{T} - \frac{1}{T_0})} \$

such that: 

\$a = \frac{1}{T_0} - \frac{1}{B}ln R_0\$

\$b = \frac{1}{B} \$

c = 0

It essentially sets the operating point around a given temperature. The further you drift away from this temperature, the less accurate the calculated temperature.  



By providing the beta at different T0, it provides the designer the opportunity to tailor their accuracy around the temperature of interest. If you need an over-temperature warning at 85C then the beta at this temperature is of interest