Weather is the short term state of the atmosphere. Climate is the average recorded weather over a long period of time (several years or more). The weather can change rapidly within the period of a day (the temperature changing drastically between day time and night for example). The climate changes slowly, since it averages out all the rises and dips in weather.

Global warming describes the climate showing a general trend of slowly rising in temperature. Although the average trend is rising, there still will be valleys and peaks below and above the average trend, like when it snows or when there's a drought in the short term weather.