Clarissa (and anyone else that cares),
  This option is available for anyone, but it typically is done by gay people who don't want to live with someone of the opposite sex, but would love to experience the joys of fatherhood (or motherhood.)  Just because their sexual appetites don't include a person of the opposite sex, that doesn't mean they don't want to be a parent.  So, yes, they take on financial, mental, moral and decisional roles, but don't live with the mother (or father.)
  And, it is supposed to work out like a conventional divorced family.  Sharing on weekends, every-other holiday, or whatever the two parties decide in advance will help the child.
  Oh, by the way--GOOD QUESTION!  That's why this forum is here, so we can ask these things, and learn.