Potential Temperature (Poisson's Equation)

Temperature alone cannot say whether two parcels of air really differ. Air that rises expands and cools, and air that sinks is compressed and warms, with no heat added or taken away. Potential temperature takes that pressure effect out. It is the temperature a parcel of dry air would have if it were brought, without gaining or losing heat, to the reference pressure $p_0$, usually 1000 hPa. A parcel that rises or sinks dry keeps the same $\theta$ while its $T$ changes, so $\theta$ is the label that tells two air masses apart at different heights.

In $\theta=T(p_0/p)^{R_d/c_p}$, $T$ is the air temperature at the pressure $p$ of the level you are looking at, and the calculation is done on the absolute scale (20 °C is 293.15 K). Where $p$ is below $p_0$, the ratio $p_0/p$ is above one, so $\theta$ comes out warmer than $T$; where $p$ equals $p_0$ the two agree. The exponent is the gas constant of dry air over its specific heat, about 0.2856. It comes from the first law of thermodynamics combined with the ideal gas law for a parcel that exchanges no heat, which is how Holton and Hakim derive it in chapter 2 of An Introduction to Dynamic Meteorology.

The page's example is air at -32.716 °C and 500 hPa, with the reference at 1000 hPa; the result is 20 °C. Read the other way, a 20 °C parcel at 1000 hPa lifted dry to 500 hPa has $T=293.15\times0.5^{0.286}=240.434$ K, which is -32.716 °C, so the thermometer reading falls with height while $\theta$ stays at 20 °C. The same relation can be rearranged for $p$ or $p_0$; fed the 20 °C and -32.716 °C pair, it returns the 500 hPa or the 1000 hPa you started with.

Read the answer by comparing it between heights. If $\theta$ rises with height, the air is stable; if it falls, the air overturns. Since a dry parcel carries its own $\theta$ with it, a layer whose $\theta$ rises with height resists vertical movement and a layer whose $\theta$ falls does not. The formula is for dry air and for a parcel that neither gains nor loses heat. Wikipedia gives the exponent as 0.286, while this page uses 287.0528 / 1005 = 0.28562. The two agree to 0.13 % in the exponent, which is 0.1 K on the potential temperature, so a hand calculation with 0.286 can differ by about that.