Relative humidity gives the ratio of how much moisture the air is holding to how much moisture it could hold at a given temperature.
This can be expressed in terms of vapor pressure and saturation vapor pressure:
RH = 100% x (E/Es)
where, according to an approximation of the Clausius-Clapeyron equation:
E = E0 x exp[(L/Rv) x {(1/T0) - (1/Td)}] and
Es = E0 x exp[(L/Rv) x {(1/T0) - (1/T)}]
where E0 = 0.611 kPa, (L/Rv) = 5423 K (in Kelvin, over a flat surface of water), T0 = 273 K (Kelvin)
and T is temperature (in Kelvin), and Td is dew point temperature (also in Kelvin).
So, if you know the temperature, you can solve for Es, and substitute the equation for E into the expression for relative humidity and solve for Td (dew point).
If you are interested in a simpler calculation that gives an approximation of dew point temperature if you know the observed temperature and relative humidity, the following formula was proposed in a 2005 article by Mark G. Lawrence in the Bulletin of the American Meteorological Society:
Td = T - ((100 - RH)/5.)
where Td is dew point temperature (in degrees Celsius), T is observed temperature (in degrees Celsius), and RH is relative humidity (in percent). Apparently this relationship is fairly accurate for relative humidity values above 50%.
More details can be found in the article:
Lawrence, Mark G., 2005: The relationship between relative humidity and the dewpoint temperature in moist air: A simple conversion and applications. Bull. Amer. Meteor. Soc., 86, 225-233. doi: http;//dx.doi.org/10.1175/BAMS-86-2-225
-- Michael Bell