Ozone is a molecule made up of three oxygen atoms. It is one of the two forms of oxygen that occurs commonly in Earth atmosphere (the other is diatomic oxygen, O2):
At low levels of Earth’s atmosphere, ozone is a pollutant with serious effects on the respiratory systems of animals, but in the upper atmosphere it plays a crucial role in preventing harmful ultraviolet radiation from reaching the Earth’s surface. This is the ozone layer.

Ozone in the atmosphere affects climate because of its interaction with energy coming from the sun. This incoming energy is spread among different wavelengths of light. We can see light that is in the visible range of the spectrum, but we cannot see the others, like ultraviolet or infrared light. All the wavelengths, including visible light, have energy (and thus heat) associated with them.
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