Commonwealth countries are the countries that were that had once been a part of the British empire and have become independent states. High commission and embassy are both diplomatic missions representing their country by another country. The high commission handles the matters involving the organization of diplomatic relations with other countries and must bring forth the purpose of one commonwealth country to another.
An embassy, on the other hand, is a diplomatic mission of a commonwealth country. However, an embassy also houses the building where the diplomats work or live. An Embassy is the official residence or offices of an ambassador. The person who is connected with an embassy is called the Ambassador. High commissions are an unspoken diplomatic contract between commonwealth countries. For example, the Indian embassy in Britain is known as the Indian High Commission.