The father of an infant who is not married to or in an exclusive relationship with the mother.
This is a Jamaican phrase which has been adopted into wider UK usage via the Jamaican community in England. It is known to be in use in the West Indies since the early 20th century. A society's need for and adoption of such a term says something about the attitudes toward marriage, i.e. it implies a society with a significant proportion of single mothers. The first evidence I can find of it in print is in a court report in the Kingston newspaper The Gleaner, in July 1932:
"I was returning from my baby father's house."
It began to be used widely in the UK in the 1990s, although it is still most commonly used in the black community.
The Independent had an article about teenage pregnancy in October 1993, which included this:
Tiana [Green, aged 17] said she wanted to have more children. Her dream was to get married to her "baby-father" and to have babies "in the wedlock".
The BBC screened a drama series called 'Baby Father' in 2002. That title was something of a play on words as the absent fathers were portrayed in the series as irresponsible and babyish.
Of course, where there are fathers there are also mothers. The term 'baby mother' is also used but is less common. This isn't quite the mirror image of 'baby father' as it is invariably the mother who takes day-to-day care of the children.