When a Jewish man marries a Jewish woman, they are most likely (and hopefully) aligned in their Jewish commitments. When she gives birth to a boy, they will schedule their son's bris. Sometimes the woman calls me, sometimes the man calls me.
When there is any other arrangement of parents, things can get a little complicated.
Father is not Jewish
When the woman is Jewish and the man is not, Jewish law has declared that the baby's Jewishness is transferred through his mother. So while the case may not look the same as the first case described above, the baby's Jewishness is not subject to scrutiny. The ceremony will look a little different because the child does not have a Jewish father, but insofar as the child being considered Jewish, this is unquestionable and unchallenge-able.
Here is where the topic of how to name the baby under that circumstance is raised. The particular question is relevant because we always name a Jewish child by his name 'ben' (son of) (his father's Jewish name). In this case, since the father is not Jewish and doesn't have a Jewish name, we would want the child to not seem different from others, so the options of how to name him are included here.
Father Not Present
While this is not necessarily a unique "family" situation, the absence of the father does raise a minor issue of who says the blessing the father usually says at the time of the Bris. You can read that discussion here
Mother Not Jewish
