When you step outside of your front door, you encounter the weather. Weather encompasses everything from temperature and precipitation, to extreme events such as tropical cyclones and tornadoes. The different between weather and climate is really just a measure of time. Weather is what's happening over a short period of time, and climate is how the atmosphere acts over long periods of time, usually around 30 years.
To study climate, there are many processes and interactions that occur in the earth system that we must understand. Many of these processes are related to two primary aspects of climate: temperature and water. The temperature is affected by the radiation from the sun and how that radiation interacts with the atmosphere and land surface. Radiation (in the form of heat) is also emitted by the land surface and interacts with gases in the atmosphere that also affect temperature. Carbon dioxide is an important gas in the atmosphere for climate because it absorbs radiation that has been emitted from the land surface, warming the air temperature. The water cycle depends on precipitation from clouds to replenish water on the ground, and on evaporation from the land surface and evapotranspiration from plants to put water back into the atmosphere to make the clouds and precipitation.
The reason it is important to study climate and how it is changing is that it will affect people all around the world. An increase in global temperature could cause sea levels to rise and change precipitation and other local climate patterns. Regionally, a changing climate can alter forests, crops, water supply, human and animal health, as well as ecosystems.
Scientists need observations of temperature and precipitation from all over the world to gain an accurate picture of the current weather. As these observations accumulate over time, a better understanding of the climate can be found.