The original atmosphere of the Earth was composed of mainly hydrogen and helium.
These gases are not very dense - they soon escaped Earth's gravity.

High level of volcanic activity at that time.
These explosions caused a build up of water vapour, carbon dioxide and ammonia in the atmosphere.
Oceans formed through condensation of this water vapour and from the ice of comets that were hitting the Earth at the time.
Carbon dioxide dissolved in the oceans.

Simple plant life and algae developed in the oceans that could use carbon dioxide and sunlight to make food.
These released oxygen into the atmosphere.
Oxygen allowed animal life to evolve.

Poisonous ammonia was broken down by sunlight into nitrogen and hydrogen.

Low density hydrogen escaped the atmosphere but the nitrogen remained.

This is only one of many theories around how life evolved on Earth. This particular theory (that life begin from some combination of energy and atmopsheric gases which formed simple organic molecules in the oceans) is known as the Primordial Soup Theory.

A famous experiment known as the Miller-Urey experiment, showed that simple organic molecules such as amino acids could have spontaneously formed from a mixture of hydrogen, methane and ammonia. However, how these molecules could have led to the formation of living organisms has never been definitively proven.