What conditions lead to the formation of precipitation?
Water and condensation nuclei.
In order for precipitation to form you first need liquid water or ice. In order to have this in the atmosphere you have to first have 100% relative humidity (or close to it). Relative humidity is the percentage of water vapour in the air as a percentage of the amount of water vapour that that air can hold at it's present temperature. The closer it gets to 100% the higher the chance that some of the water vapour will change back to liquid (or solid) water.
The problem is that water doesn't like to change state readily. It usually needs a little "kick", that is where the condensation nuclei come into play.
Condensation nuclei are particles of dust or other things that the vapour can actually condense around (hence the name). Imagine a steam room, the liquid water is more on the walls and the ceiling than it is in the air. That is how condensation nuclei work.
Once you have both of these then all you need is the liquid drop to become large enough so that gravity on it overcomes the updrafts of the atmosphere. Then it falls, tada, precipitation.