Adi Shankaracharya was the most prominent teacher of the Advaita school of Vedanta. His commentaries to the Srimad Bhagwat Gita, the Upanishads, and the Brahma Sutras define the parameters of Advaita thought. Although it must be kept in mind that all the Vedanta philosophies go back to the Upanishads.
Adi Shankaracharya is revered as the pre-eminent preacher who continued the Advaita tradition and "Shankaracharya" has become the title for the heads of the many Advaita schools in India today. This is because of the great fame and respect associated with it.
The philosophy of Advaita literally means 'non-dualism'. It is the oldest extant among the Vedanta schools of Indian philosophy. The unending quest, first begun in the Upanishads, is to decipher "brahma", the eternal source of the universe; the Atma (higher-self) and the Self (ego), and the mutual relationship between Atma and Brahma.
The Upanishads look at the issues from all the different angles but the school of Advaita teaches essential identity between the Atma and Brahma. In the other Vedantic practices, this relationship between Brahma and Atma is understood in many different ways.
The swan has a very special association with Advaita. The swan is also called 'Hansa' in the Sanskrit language and all of the greatest teachers in Advaita Vedanta are called the 'Paramahamsas' - the eternal swans. The word 'Hansa' is the full form of 'Ham' meaning 'I am He'. It represents the state of highest realisation. A swan stays in water for the entire day, but interestingly, its feathers are never wet. In the same context, an Advaitin lives in this materialistic world and yet remains unaltered by the life's ups and downs. The Swan is also believed (mythically) to possess the ability to separate milk from a mixture of milk and water. The Advaitin also differentiates the Atma from the materialistic world. The Atma is here bound up with the world. It is like milk, which cannot be separated from a mixture of milk and water. It can only be realised with the right discrimination between the ephemeral and the eternal. This is important for the Advaitin. Thus the swan is the symbolisation of the Moksha, ie., liberation while still alive by the realisation of Brahma.