How was the atmosphere of the Earth formed?

In the very early phase of the Earth life the planet was a mass of melted rock that progressively cooled down.

During this process gases escaped from the rocks accumulating to form a primordial atmosphere likely similar to the one of Venus.

Further gases, including water vapor, were emitted, and are still emitted, during volcanic activity that, in the early stages of the planet, was much more intense than now.

The atmosphere was depleted of oxygen and this gas started to accumulate only when photosynthetic life forms evolved on the planet. Other geological events, such as changes in the redox state of the gases emitted from the mantle of the planet, also contributed in the rising of oxygen concentration.

This event is called "The Great Oxidation Event" and happen around 2.4 - 2.5 billion years ago leading to the extinction of many of the lifeforms that lived in an anoxic world and opened the path for the evolution of the life of oxygen-breathing organisms.

In the 1960s, Heinrich Holland of Princeton came to the conclusion that gases emitted by Earth's volcanoes were the source of the planet's primordial atmosphere page 447–477 Model for the Evolution of the Earth's Atmosphere.

Abelson Philip H. "Chemical Events on the Primitive Earth" National Academy of Sciences USA 55 1966 pages 1365–1372) states that the water released from the volcanoes underwent a process called photodissociation, which splits water molecules into hydrogen and oxygen. Since hydrogen is lighter than oxygen (mass 2 for hydrogen, mass 32 for oxygen), a large portion of the hydrogen would disperse into space, leaving an atmosphere rich in oxygen.

Because of the water that volcanoes release, there was never a period of time when oxygen made up a sizable portion of the earth's early atmosphere.

In the early history of the planet, water vapor from comets and asteroids, as well as gases released by volcanic activity, all contributed to the formation of the Earth's atmosphere.