British subjects are now a residual group of British nationals. They are, as defined in the 1981 Nationality Act, those citizens of the United Kingdom and Colonies (as defined in earlier Acts) who did not become British citizens or citizens of any other country. In the main, they are people born before 1949 (mostly in the Republic of Ireland or what was hitherto India) who, in the case of those born in the Republic of Ireland, have applied after 1949 for restoration of their British subject status, or who, in the case of those born in India, did not acquire citizenship of their country or of any other Dominion.
Some British subjects (principally those connected with the Republic of Ireland) have a right of abode in the UK, others do not. British subjects with the right of abode in the UK are considered UK nationals for EU purposes, and their British passports are in the EU format. Other British subject passports do not contain any references to the EU. British subjects are not, however, British
citizens and they do not benefit from, for example, access to the US visa waiver scheme or the Australian ETA scheme in the way that British citizens do (although this is the choice of the USA and Australia, not of the UK, of course).
It is all more complicated than what I have said above, and the term British subject has had different meanings over the past century - see for example, this
article on Wikipedia. There are in fact a number of types of British national - see for example, this
article on Wikipedia.
Note that people often erroneously refer to the British people as "British subjects", citing the fact that they are subjects of the Queen and so on and stating that they are not citizens. In fact, they have been citizens since 1949 (Citizens of the United Kingdom and Colonies and then British Citizens), and they are no longer, since 1983, "British subjects".
