Weathering is an interesting word. Technically, it refers to the wear and tear upon an object as a result of existing in an atmosphere. But you can also refer to mechanical weathering, covering basically any wear-and-tear that is not the result of the weather. If you were to refer to the moon as a well-weathered astral object, few would criticize your usage -- the craters constitute weathering, even though you could hardly find a place with less weather.
I will be writing about only one type of weathering in this write up -- specifically, the results of atmospheric activity on rocks. Weathering is oft referred to in the field of geology, but the meaning isn't always clear. Let's see if we can fix that.
Wind erosion
Wind doesn't have any direct erosional power on rocks in of itself, but when it picks up sand and other small particles it can do some real damage. Wind can sand-blast rock and rock formations to a smooth polish, making beautiful curves and sharp edges, and making grooved, pitted or fluted surfaces. While it makes some impressive rock formations, it isn't actually a major influence on the average landscape; other types of weathering can do more, and do it faster, and the conditions under which sand and wind are both present in great enough quantities to make distinct formations are comparatively rare.
If you would like to see some examples, perhaps one of the most impressive is the Upper Antelope Canyon (AKA Tse Bighanilini’), a highly eroded canyon with some of the most beautiful cliffs you will ever see. White Sands National Monument, in New Mexico has some impressive windscapes, spotted with formations called yardangs. Sandstone gives rise to a number of impressive formations, such as The Wave and other rather psychedelic formations in Utah and Arizona. Despite most of my examples being in the American mid-west, such formations can be found the world over, particularly near deserts.
Wind erosion is best known to us humans by its effect not on rocks, but on dirt, through a process known as deflation. This is simply the wind picking up dust, silt, sand, and dirt, and moving it from one place to another. Unfortunately, this often involves picking dust off of a farmer's field in time of drought, and dropping it in the ocean or other non-agricultural area, as happened in the Dust Bowl, and as is still happening all over the world today.
Water erosion
This is the big one -- water erosion makes some of the most impressive formations on Earth. Witness the Grand Canyon and the Carlsbad Caverns. And the most important form of water erosion is simply rain. Thus starts the cycle that carries water inland to form streams, rivers, and lakes. The rain collects on rocks, changing their chemical composition (see chemical decomposition, below), freezing and expanding (see ice erosion, below), and simply washing away tons of dirt and rock every year.
Water also causes mud-slides and avalanches, and slowly carves away dirt and rock, making arroyos and canyons. It does this both by abrasion (dragging sand and gravel across stream and river beds) and also by absorbing minerals from the rock, wearing holes in limestone and slowly breaking down harder rocks.
At the shore, waves and tides can also cause significant weathering. Examples of tidal weathering are less impressive than other sorts of weathering, as it usually consists of simply moving the shoreline and little else. In some cases sea water will eat into the base of a limestone cliff, making caves or even a series of caverns. These are sometimes called napes. The Hopewell Rocks are a marginally interesting tidal formation, in the Bay of Fundy at Hopewell Cape, New Brunswick, Canada, in which extreme tides have washed away much of the shore, leaving sculpted rock towers. Sea stacks such as the Yesnaby Castle in Scotland are an example of weathering through the action of sea waves.
As with wind erosion, when humans think of water erosion, they usually don't think about rock, but soil. Water washes away a tremendous amount of soil every year, especially from areas in which humans have disturbed the natural ground cover (for example, farmer's fields). Iceland is an excellent example of why one should not trust water. Not quite as extreme (not yet, anyway) is the more famous example of the Yellow River in China, which has carried off billions of tons of topsoil, degrading the land and damaging the ecology of the river.
Ice erosion
While I called water erosion the big one (and it is), an amazing amount of the geology you see every day is a result of glacial erosion. At one point or another, most of the Earth had a glacier or ice sheet running over it, crushing the land, wearing down the water-made V-shaped valleys into flattened U-shaped valleys, cutting into peaks and making them sharper, as with the Matterhorn. The material cut away was deposited into moraines, resulting in low hills and mountains, sometimes on an enormous scale -- Long Island is a glacial moraine.
Ice sheets created large shallow basins with their weight, making lakes. These include hundreds of lakes throughout North America, including the Great Lakes. Glaciers also left behind thousand of smaller lakes and pools found throughout the world.
One of the biggest effects of glaciers and ice sheets is that they pick up and grind rocks as they move along. They break rocks off of mountains, they grind rocks into gravel, and they grind gravel into sand and dust, including an extra fine dust called rock flour or loess. As they move along, and again as they melt, they help move these rocks far from their point of origin.
While ice seems rather innocuous in todays world, on a geologic scale it has been a major player. Today it still plays an active roll, as water seeps into cracks in the rock, freezes, expands, and cracks rocks apart (AKA frost wedging). Even so, now that the Earth is going through a warm period, ice and snow are probably most often touted as a good preventer of erosion, keeping water locked up in snow packs, and releasing it slowly throughout the year.
Chemical decomposition and disintegration
No, I'm not referring chemical spills or pollution. The chemicals in this case are natural, and include oxygen, carbon dioxide, and water. While not as impressive as other forms of weathering, the chemical decay of rocks is going on constantly, causing the stones exposed to the air or water to slowly crumble. This gives the other forms of erosion a strong leg up, and while it is not an obvious form of weathering, it is a very important one. In fact, when a geologist refers to 'weathering' rather than 'erosion', they are often referring to chemical weathering alone.
The most familiar example of this is iron, which will react with oxygen and rust. Iron is not uncommon in the Earth's crust; it appears in the mica in granite, and in the pyroxene in basalt. When exposed to the elements, these slowly turn to limonite (essentially a mineral form of rust). This is only one of the chemical reactions that can weaken rock, and when these reactions disrupt the grain of the rock, they allow moisture to seep in. This water washes away further minerals, and when carbon dioxide combines with the water it forms carbonic acid, which works to convert more minerals into forms that are soluble in water.
This is the process by which limestone is eroded, resulting in impressive caves and speleothems. Feldspar is also a common victim of chemical decomposition, turning slowly to clay. One mineral that doesn't decay much at all is quartz, which remains after other minerals are gone, often in the form of sand, which in turn weathers rock. Note that while the quartz may break into smaller pieces (undergoing disintegration), it does not change its composition (decomposition).
One interesting feature of chemical decomposition is a process known as exfoliation, in which chemical changes happening under the surface of the stone cause minerals in the interior to expand, cracking the rock. One way this can happen is if water dissolves salt, is drawn into microscopic cracks or capillaries in the stone, and then dries. The salt crystallizes and expands, causing the rock to shed small chips from its surface.
Other forms of weathering
Biological activity is often considered to be a type of weathering. This consists mostly of plant activity, such as trees and shrubs breaking up rock with their roots. This may be referred to as 'mechanical weathering' to distinguish them from atmospheric processes.
Fire: The heat from forest fires can cause rocks to crack and flake. the surface of the rock will heat up much faster than the deeper rock, and flakes will pop off of the surface of the rock. This is a very minor source of weathering.
Gravity: Things fall down. This includes mountains, albeit slowly. Gravity is usually aided by one or more of the weathering forces mentioned above. The slow downward movement of rocks and soil is called mass wasting. While it is often hard to to separate the effects of gravity from the wear caused by water, air, and ice, those effects clearly due to gravity are generally not considered weathering.
Factors that effect weathering
Weathering is usually the most severe in warm, wet climates. Obviously, since running water is such a great eroder, more of it means more erosion. Water and warmth also help to speed up some of the chemical decomposition. Another important factor is that warmer, wetter climates are also the climates that have the most plant growth. Not only do plant roots break up rock, but some acids that form in humus and other decaying plants increase chemical decomposition. Wet climates will generally have more chemical decomposition, while drier climates will be largely limited to disintegration.
Weathering will also usually be most sever on steep slopes; this is largely because gravity is helping, both helping to pull down rocks, and clearing debris so that fresh rock is exposed to the elements.
Steep slopes are often accompanied by high altitudes, and thus frost. Areas that undergo freezing and thawing cycles will erode more quickly than those that are permanently frozen.
But weathering is not...
By now you may be getting the impression that anything and everything constitutes as a form of weathering. As I said at the start, weathering is not a very well defined term, but there are some exceptions. Generally speaking, volcanic activity, the formation of rocks (sedimentary and otherwise), the effects of earthquakes and tectonic plates shifting, and meteor impacts should not be considered examples of weathering. Human activities would not usually be considered weathering.
Despite my peripheral mention of such things above, deposits such as stalagmites and moraines might not be considered proper examples of weathering -- they are deposits of new formations, not destruction of old ones.