The climate is the weather averaged over a long period of time. A descriptive saying is that "climate is what you expect, weather is what you get". The IPCC glossary definition is:
Climate in a narrow sense is usually defined as the “average weather”, or more rigorously, as the statistical description in terms of the mean and variability of relevant quantities over a period of time ranging from months to thousands or millions of years. The classical period is 30 years, as defined by the World Meteorological Organization (WMO). These quantities are most often surface variables such as temperature, precipitation, and wind. Climate in a wider sense is the state, including a statistical description, of the climate system.
The exact boundaries of what is climate and what is weather are not well defined and depend on the application. For example, in some senses an individual El Niño event could be considered climate; in others, as weather.
When the original conception of climate as a long-term average came to be considered, perhaps towards the end of the 19th century, the idea of climate change was not current, and a 30 year average seemed reasonable (but see note 1). Nowadays, knowing that there are long-term trends in the temperature record, it is harder to give a precise contradiction-free definition of climate: over a 30 year period, averages may shift; over a shorter period, the statistics are less stable.
In a given geographical region, the climate generally does not vary over time on the scale of a human life span. However, over geological time, climate can vary considerably for a given place on the earth. For example, Scandinavia has been through a number of ice ages over hundreds of thousands of years (the last one ending about 10,000 years ago). Paleoclimatology is the study of these past climates.
In the original sense, climate is a concept used to divide the world into regions sharing similar climatic parameters. Climate regions can be classified on the basis of temperature and precipitation alone. Examples of such climate schemes are the Köppen climate classification or the Thornthwaite climate classification schemes.
For more details about specific climates, please see: