It is unknown what the source of the original atmosphere was. The oceans were believed to have formed from this original atmosphere. The original atmosphere could have formed fromed with the rest of the planet, from accumulated debris. It could have been released from the ineterior of the earth through volcanic activity. It could have been deposited by comets and icy particles.
Weather refers to the circulation of the Earth's atmosphere.
Sunlight heats the Earth and the atmosphere tries to redistribute the heat from warm areas to cool areas. The atmosphere responds to the change heat input of the Sun. Climate refers to atmospheric changes that occur over decades and centuries. The beginning and ends of ice ages refer to changes in climate. It appears as though the chemical makeup of the atmosphere has been changed by the presence of life on Earth.