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For comparison, read about the formation of Jupiter's atmosphere.
Earth's Primordial Atmosphere
Scientists ask many questions. One of the questions they like to ask is "Where did the atmosphere come from?" As always, scientists chip in with many different, and sometimes conflicting answers. Some scientists say that there was no atmosphere to start with, that the Sun, in its T-Tauri phase blew away the atmosphere with a strong wind. Other scientists point to evidence that comets may have brought water vapor to Earth and the other terrestrial planets. Different scientists think that the very first atmosphere of the Earth must have been made of the same material as the sun, stars, and the planet Jupiter; with the same amount of each element as those bodies, scaled to the size of the Earth. (Obviously Earth can't have as much hydrogen as the Sun or it would look like the Sun! But it can have the same balance of hydrogen to oxygen as that in the Sun). The amounts of each gas in such an atmosphere scientists call "cosmic amounts". They say that it was too hot for gases to condense or rain to the ground, so there was no ocean. One thing they seem to agree upon however, is that the atmosphere we started with is nothing like the atmosphere we have today. You may ask, "How do they know that?"Well, one way to tell is by measuring an element which doesn't change. That is, measure an element which is to too heavy to escape from Earth so it doesn't go anywhere, and doesn't react with anything so it doesn't change to a different form and hide from sight. One such element is 20Ne (Neon). It hasn't escaped and it doens't change into something else. What is here is what has always been here. Scientists have measured 3 things; the amount of 20Ne in the atmosphere today, the amount of nitrogen in the atmosphere of Earth as well as in the Sun. If Earth still had its original atmosphere, then the balance of the amount of nitrogen to the amount of neon in the Earth's atmosphere would be the same as that of the Sun, namely "cosmic" amounts. What they learned from these measurements is that the amount of nitrogen compared to the amount of Neon in the Earth is 10,000 more that it should be! This means that all the extra nitrogen in the Earth's atmosphere is new and must have come from somewhere. Moreover, with small planets like the Earth, Mars and Venus the atmosphere will eventually drift away unless it is replenished somehow. Scientists call the new atmosphere a secondary atmosphere, and they think all the new gases came from volcanoes. This suggests that, with Earth at least, the atmosphere is continually replenished by volcanic activity.