Frost is deposition of water vapor directly into ice crystals on a cold surface.
Water molecules in the air are constantly bouncing off of surrounding surfaces. If a surface is cold enough, the molecules will stick to the surface rather than bouncing off. Once a layer of frozen molecules form, new molecules will tend to align themselves with those already there, forming regular crystals.
The temperature of the surface is determined by the net heat balance on it, which is a function of air temperature, air movement, radiation, and solid conduction. Usually the ground is warmer than the air, and the air tends to be the strongest effect, but a surface isolated from the ground on a clear, still night will be dominated by radiation to the sky, so it can get colder than the air. That's why you see frost on top of the mailbox or on the tips of the grass when there is none on the ground.