In Vedic era a Brahmin was a person who had attained Self- knowledge or Atma Jnana. This was an extremely difficult path of discipline of body, mind , and intellect, and people irrespective of their birth or class, who dedicated to such an austere life were recognized as Brahmins.
A great example of this tradition (that a person becoming a Brahmin, rather than born as one) is the case of Vishwamitra, a warrior (kshatriya), who became a Brahmin after attaining Atma Jnana or Self-Knowledge.
A smritis, or code of conduct composed by sage Atri defines brahminhood very clearly.
"By birth, every man is a Shudra (an ignorant person). Through various types of disciplines (samskaras), he becomes a dwija (twice born). Through the studies of scriptures, he becomes a vipra (or a scholar). Through realization of supreme spirit (brahmajnana), he becomes a brahmin."
The belief that people born in Brahmin caste, automatically become Brahmins, is a much later concept in the very ancient India. Thus, Brahmin means not caste but one who has attained Atma Jnana or Bramha Jnana.