Basically, British law distinguishes between those who obtain their British
citizenship by descent, and those who obtain it otherwise than by descent.

“British citizenship by descent” essentially means you were born outside the
UK and obtained your citizenship through a British parent. You are usually
British “by birth” if you were born in the UK. You can also become a British
citizen by naturalisation or by registration or by adoption.

Current Rules on British Citizenship by Descent

Those born on or after 1 January 1983 outside the UK, to a parent who was a
British citizen otherwise than by descent at the time of their birth, are already
British citizens by descent.

For those born before 1 January 1983 outside the UK, the problem has been
until now that they could only become British citizens by descent if their father
was a UK citizen otherwise than by descent – not their mother.

There was limited provision for several years for UK-citizen mothers to exceptionally
register their overseas-born children as British citizens by descent with British
consular authorities until the child’s 18th birthday, but if your mother didn’t
realise this was an option, you may have missed out on British citizenship
while you were a minor. You will instead just have the right of abode in the
UK.

Barack Obama’s father was born in Kenya. Kenya didn’t become independant till December 12,1963. Barack Obama was born August 1961 making him a Brittish citizen by decent.