The term 'British subject' has had various meanings over the years. However, under UK law, it now solely refers to a small number of people who were born before 1949 in the Irish Republic, India and Pakistan (and who did not acquire citizenship of those countries).
See here:
http://en.wikipedia.org/wiki/British_subject
If you've come across the phrase 'British subject' in the visa requirements of another country, it may refer to those people who are now known as 'British dependent territories citizens' and/or 'British overseas citizens'. Such people are officially 'British' but they don't have the right to live in the UK.
Chris