Weather refers to the instantaneous or near-term state of the atmosphere in a given location. Put simply, weather is what the sky is doing right now. More generally, weather refers to what variables such as temperature, precipitation, and wind are doing over the course of seconds to hours, days, or even a week. Phenomena such as snowstorms, tornadoes, the passage of a warm/cold front, etc. are all examples of "weather." Weather can vary quite rapidly in a given location, in sharp contrast to climate which takes years to vary (because it's a long average). Unlike climate, weather is typically hard to predict accurately for a week into the future (sometimes predictions are hard even a few days out). See climate vs weather for more details.
In a related but somewhat different context, weather may also refer to unforced seasonal or yearly variability associated with natural changes in the atmosphere-ocean system, which is to be distinguished from forced changes by factors external to the climate system.
See also the AMS definition of weather.
People typically encounter mild weather in their daily lives, depending on the season and their location. However there are extreme weather events that occur in every location that are usually unexpected for their location's history, and can often be disastrous. In most cases, extreme events are defined as lying in the outermost 10% of a location's history. Types of extreme weather include hurricanes, tornadoes, blizzards, and droughts. Some of these events can be seen below.
Figure 3: Hurricanes are giant storms that form over warm bodies of water and can devastate entire cities.
Figure 4: Blizzards leave entire cities under several meters of snow in many cases.
Figure 5: Droughts can last for months to years, impacting ecosystems and agriculture, and damaging economies.
For more information on extreme weather, click here.