When air rises, it cools due to expansion. That's because air pressure results from the weight of hate overlying air. The higher you go, the less air is above you, so the pressure drops. And as pressure drops the air expands. Just as a spray can cools, so the air cools in updrafts.
Rising air cools until it reaches its dew point temperature. This is the temperature sat which water vapor in the air starts to condense. At the surface, this is called fog. You've felt fog droplets against your face and hair. Aloft, it's called a cloud.
The condensation of liquid water droplets releases the same amount of heat that was required to evaporate that water from the oceans in the first place. This latent ("hidden") heat of condensation warms the air and makes it more buoyant, so it rises even faster. In the middle of a thunderstorm, updrafts can reach 100 mph due to the "rocket fuel" of condensing raindrops!
Virtually all rain and snowfall is associated with rising air. Locally, this can result from thermals rising from the sun-warmed ground. At larger scales, it's almost always due to air being pushed over an obstacle: either another airmass or a mountain range.
In the tropics, the Trade Winds from each hemisphere collide in the Intertropical Convergence Zone (ITCZ) and rise strongly. This causes a band of very heavy rain along the Equator that shifts north in July and South in January. On land, the band of heavy rain associated with the ITCZ is responsible for rainforests in the Amazon, the Congo, and the Maritime Continent. In the ocean, the ITCZ often spawns tropical cyclones, which are known in the Atlantic as hurricanes and in the Pacific & Indian Ocean as Typhoons.
Adjacent to the tropics, the poleward moving air has had all the water vapor wrung out of it by tropical rainfall. This dry air slowly sinks in the subtropics. As it falls it compresses and warms by the same process that rising air cools. Compressive warming evaporates every microscopic cloud droplet so that the subtropics are clear and sunny and very dry. These are Earth's desert belts: home to the Sahara, the Arabian desert, the arid stretches of Central Asia and Australia and Southern Africa.
In the middle latitudes of both hemispheres, fronts sweep warm moist air up and over cold dense air, wringing out the rain and snow. These warm and cold fronts are waves that form in the Prevailing Westerly winds, shaking the jet stream to the north and south like a snake. The westerlies wrap around the winter pole as a great Polar Vortex whose spin prevents the tropical warmth from reaching the frigid latitudes. Waves in the polar vortex grow at the expense of the average flow, sweeping cold polar air equatorward in winter blasts and allowing subtropical warmth to penetrate poleward.
Tropical rainfall belts irrigated by the ITCZ are home to evergreen broadleaf forests, the most productive and diverse ecosystems on Earth. Millions of plant and animal species, along with fungi and microbes, use all that water to cycle oxygen to carbon dioxide and back again with the energy of the Sun. Middle latitudes also host productive forests, but many are deciduous, jettisoning the machinery of photosynthesis each winter to regrow it back in springtime. Earth's forest belts are bounded by grasslands and savannas that grade into desert in the subtropics.
