The definition of precipitation is a release of water from the atmosphere to reach the surface of the earth. Precipitation can be in any form, covers all forms of water being released by the atmosphere, include rainfall, snow, drizzle, sleet (combination of snow and rain), and hail (balls of ice with 5 to 125 mm in diameter).
The precipitation it self is related with water cycle in our earth systems. The water at the atmosphere are supplied by a process called evaporation (a process of liquid water change into vapor and transport to the atmosphere) from the ocean and several part from water surface on land. The ocean which is stored about 96.5 percent of earth water is supplied as much as four times much precipitation which occur on land. In the other hand, precipitation in the ocean is almost four times higher than on land.
During rainfall process, water is evaporated, forming a cloud at atmosphere, when it saturated enough, the falling water drops from the atmosphere to the earth surface, infiltrates to the soil. The soil possesses water storage capacity according to its properties, such as soil types and grain size. When rainfall lasts longer, soil can be saturated and prevent water to infiltrate. Excess of water may inundate soil surface in a large scale will cause a flood and in the case of sloping surface, water may flow as surface runoff.
The precipitation is measured in units of depth of fall (rainfall depth) per unit of time, known as rainfall intensity (rainfall amount per time interval) expressed in mm in meter square per day. For example : 1 mm per day rainfall is a quantity to cover 1 meter square ground to a depth of 1 mm in 1 day that none is lost by runoff, evaporation, or sinking into the ground.
At least there are two types of precipitation in rainfall form: Convective rainfall and Orographic rainfall. Convective Rainfall occurs as a result of adiabatic cooling associated with parcels of air that rise because they are less dense than the air that surrounds them or warm air up to a cool, dense environment. Orographic Rainfall precipitation increases with elevation, this happens in mountain area where the higher altitude of area, then higher level of rainfall which occur.
For more info you can download the Lecture of Hydrology about Precipitation in reference link below.