Climate and weather are both components of meteorology.

In brief: the weather of any place refers to the atmospheric variables for a brief period of time. Climate, however, represents the atmospheric conditions for a long period of time, and generally refers to the normal or mean course of the weather. Climate can now be expanded to include the future expectation of long term weather, in the order of weeks, months or years ahead.

A more detailed definition from the Meteorological Glossary published by the U.K. Meteorological Office is:

The climate of a locality is the synthesis of the day-to-day values of the meteorological elements that affect the locality.

Synthesis here implies more than simple averaging. Various methods are used to represent climate, e.g. both average and extreme values, frequencies of values within stated ranges, frequencies of weather types with associated values of elements. The main climatic elements are precipitation, temperature, humidity, sunshine, wind velocity, and such phenomena as fog, frost, thunder, gale; cloudiness, grass minimum temperature, and soil temperature at various depths may also be included. Climatic data are usually expressed in terms of an individual calendar month or season and are determined over a period (usually about 30 years) long enough to ensure that representative values for the month or season are obtained.