Weather is defined by meteorologists as the state of the atmosphere at a given point in time. Typically, they are only referring to the troposphere, the lowest layer of the atmosphere. The weather is determined by 7 main factors: air temperature, air pressure, cloud type, precipitation type, precipitation amount, wind speed, and wind direction. As an example, the full weather report of where I live right now is: 29.8o F; 30.02 in pressure; cumulus clouds; light snow; up to 2 in; 1.0 mph winds, from the south-east.
Weather vs. Climate
Weather is not the same thing as climate. This is a really common misconception with all the news about "global warming" and "climate change." Weather is local and short term. It is snowing here right now, but it was sunny yesterday. Also, it's probably not snowing all over the Midwest, just here. Climate, on the other hand, is long-term and broad, covering regions to the entire planet. Thus, on average, over much of the planet, annual temperatures over the past 100 years have been getting warmer. A freak snowstorm does not mean climate change is not occurring, nor is it necessarily caused by climate change. The two are related, but are not directly connected. So, as much as people like to claim that this bitter cold winter disproves global warming, that's simply not how it works.
That's basically all there is to weather. It's a local set of atmospheric conditions. For sounding so simple, and even boring, however, there's a lot of unpredictability in weather, which can lead to some pretty terrifying phenomena.
More on weather.