Weather refers to any type of weather effect in DF. As of this version, it will be either be rain or snow. Rain cleans any tile it lands on, removing blood, vomit and other bodily fluids on contact. Any dwarf caught outside when it rains will receive a minor unhappy thought.
When rain hits a tile labeled as a murky pool, it will begin to fill it up with 1/7 water, and if that does not evaporate the water will grow deeper, until the pool is full. Murky pools do not overflow from rain, but this extra water can be drained off and stored/used. (See the Well Guide.) While not much, it can really help maps without 'unlimited' water supplies such as rivers and brooks.
Freezing will also refill murky pools which are found in some biomes (mainly temperate) - after the murky pool unfreezes, it will be full again. This happens in biomes of moderate temperature as well as cold biomes.
Weather can be disabled by changing [WEATHER:YES] to [WEATHER:NO] in the init file. Disabling weather is a quick and largely harmless fix to improve framerate on older machines if required.