Dewpoint is a proxy for the concentration of water vapour molecules in the air (i.e. the
mixing ratio). The higher the dewpoint, the more molecules of water there are per kg of air (mixing ratio is quoted in grams of water per kg of air, g/kg). At standard atmospheric pressure (1013.25 hPa), the mixing ratios of different dewpoints are listed below.
Consider a tiny speck of dust/smoke/pollen (condensation nucleus). There is always an equilibrium between condensation and evaporation of water molecules in the surrounding air onto and from its surface. This equilibrium depends on the temperature of the air and the mixing ratio.
At a certain temperature (above the dewpoint) some of the molecules have enough energy to break free of the surface tension of the droplet while other do not (the energy distribution has a finite width). As the temperature changes, the equilibrium shifts, in favour of evaporation (condensation) as the temperature rises (falls). As the temperature falls, condensation is favoured until the temperature matches the dewpoint and evaporation is suppressed enough to bring the mixing ratio of the surround air down to the value for that dewpoint. Further cooling causes more condensation and hence the droplet grows, following the Clausius-Clapeyron relationship.
Take a cloudless autumn night. Radiative cooling causes solid surfaces (e.g. a blade of grass) to cool more rapidly than the overlying air (as measured at 1.5 metres). But the air immediately surrounding that surface cools to the same temperature as the surface and, depending on the dewpoint/mixing ratio of that air, may reach the dewpoint and cause condensation to occur on the surface (i.e. dew-formation). Further cooling causes the dew droplets to grow. If the dewpoint and cooling temperature of the surface are both below 0 °C then that dew will actually be frost.
It's the same for radiation fog, except in this case the cooling to saturation occurs through a much deeper layer of air (1 to hundreds of metres). Advection fog occurs when warm, moist air moves over a cold surface.