The Earth's atmosphere is the collection of gasses and water vapor (often in the form of clouds) that exists for several miles above the surface of the planet. The atmosphere is incredibly important to all weather cycles and life on the planet in general. The atmosphere helps keep heat from the Sun that makes its way to the planet's surface trapped near the surface and keeps it from escaping back into space. This keeps the surface temperatures at habitable levels, thus allowing life on the planet to exist. The atmosphere also keeps the Earth's water from escaping into space as well. The water in the atmosphere in the form of clouds and precipitation determines the weather and climate. Weather and climate are basically the same two things, just on different time scales (climate is weather patterns for a general area over a long period of time). When we talk about climate and weather, we are really talking about temperature and precipitation. The atmosphere determines both of these factors. The amount of heat and moisture that the atmosphere retains in a given geographic area will determine the weather pattern for that area as well.
The fact that we have an atmosphere at all is largely the reason why we have climate and weather; the atmosphere provides a sort of buffer that insulates the surface of the Earth from solar energy, and allows liquid water to exist at the surface. Other planetary bodies in the solar system that have atmospheres, such as Venus, Jupiter or Titan, have been observed to exhibit weather and climate effects as well, such as winter snows or stormclouds.
Perhaps the most significant determination of climate on Earth is the presence and abundance of water, and this is largely determined by the water cycle, which, in turn, is dependent upon the atmosphere to circulate the water onto land via clouds and rain.