Weather is the moment-by-moment and place-by-place description of the state of the atmosphere, with events taking place on a small scale in both space and time. Weather is the current atmospheric conditions, including temperature, rainfall, wind, cloud cover, humidity, etc. at a given place. Weather is what's happening right now. The weather can change a lot within a very short time. For example, it may rain for an hour and then become sunny and clear.
Climate, on the other hand, encompasses the totality of all weather conditions (in a given region) accumulated over a long period of time. Climate includes all of the statistics of weather observations (in a given region). By international convention, climate statistics are computed from all weather elements compiled over a 30-year period. However, depending upon what process we are studying, a much longer time period may be necessary since there are variations in climates that happen over periods much longer than 30 years. For example, if we are interested in the Ice Age cycles, we must consider time periods on the order of a million years.
Climate is sometimes referred to as "average" weather for a given area, but I argue that this is an incomplete definition of climate. The National Weather Service uses data such as temperature highs and lows and precipitation rates for the past thirty years to compile an area's "average" (also known as the "normal" of a statistical distribution) weather. However, you need more than "average" weather to accurately portray an area's climatic character - variations, patterns, and extremes must also be included. Thus, climate is the sum of all statistical weather information that helps describe a place or region, which includes averages, but also standard deviations about the average, and all other statistical information that can be computed from weather observations over an extended period of time. It is misleading to say that climate is simply "average" weather. (See this instructive figure for standard deviation.)
Some meteorologists say that "climate is what you expect and weather is what you get." For example, on any given day in January, we expect it to be rainy in Portland, Oregon, sunny and mild in Phoenix, Arizona, and cold and snowy in Buffalo, New York based on our understanding of the climates of these cities. However, the weather conditions for a specific day can be quite different -- it can be sunny and mild in Portland and rainy and cool in Phoenix for a specific day in January. An anonymous individual once said "climate tells you what clothes to buy, but weather tells you what clothes to wear." That is particularly relevant for winter in Tucson. On some days the high temperature is in the 80s and you may wear shorts, while on some days the high temperature is in the 40s and a heavy jacket is required. The climate suggests a variety of clothing may be needed, while what you wear on a given day depends on the weather for that day.
In order to describe the climate of a region, it is necessary to know the extremes as well as the averages. It is one thing to know the average annual temperature and rainfall of a place, but it is much more meaningful to know how hot it gets in the summer and how cold it gets in the winter; whether the rains tend to fall in thunderstorms or in extended drizzles; whether there are wet and dry seasons; and whether the place is subject to occasional high winds, tornadoes, or hurricanes. The extremes of weather often determine what types of plants and animals are able to live in a particular climate. Examples of important climate statistics include:
Therefore, it is not correct to define climate as the average weather for a given location. To fully describe the climate of a region, we need to know the extremes and frequencies of every weather occurrence that may be of interest. Consider the climates of Tucson, AZ and San Diego, CA. These two cities are located very close together and have similar yearly average temperatures and precipitation, but the climates are quite different. The WORD document linked at the end of this sentence, presents an interesting comparison between the climates of Tucson, AZ and San Diego, CA (see WORD document). This example shows that we need to know much more than just the annual average temperature and average precipitation to understand the climate of a region.
Perhaps the biggest misunderstanding by the general public concerning weather vs. climate is to cite extreme weather events as evidence that something is up with the climate. For example: the debate about global warming seems to get more attention during a prolonged summer heat wave; or after a disasterous severe weather event such as a tornado outbreak or a strong hurricane, you get people saying that the climate is becoming more hostile (possibly due to the evil actions of people). We need to realize that a particular extreme weather event by itself tells us nothing about climate change. Extreme weather events have been noted all through human history and will continue to occur into the future. Changes in climate take place over many years, thus for climate change, there would have to be a measurable change in the frequency of extreme events over a prolonged period.
Something else that you need to understand is that climate change is a normal and natural part of Earth's history. We know that climate change has occurred throughout the history of the Earth. We tend to think of today's climate as "the way things have always been" or "the way things ought to be", but this is incorrect. There is overwhelming evidence that climates on Earth are always changing.
We can explain some of the reasons for past climate changes (for example, ice age cycles are probably related to slight variaitions in the Earth's orbit about the Sun), but we certainly cannot explain everything because the climate system is so complex. We also know that past climate changes have had a significant impact on the distribution and evolution of the various forms of life on the planet. This works both ways though because life itself also helps to determine climate.
Climate changes have also had a significant influence on the spread and developement of human civilizations. For example, in the 13th century the Vikings established agricultural settlements on Greenland, which were based on farming and sheep herding. After 100 years or so, the climate of the region became colder, sea ice and glaciers spread over the area, and the settlers died or abandoned the area.