The title is a play on the 1950 film Father of the Bride (or its 1991 and 1995 remakes).
Sylvester finds a sparrow egg in a nest. Before he can eat it for breakfast however, a little bird hatches out and calls him "Mama." He warms to the idea of being a father and instead of eating it, he protects the bird throughout a series of near-disasters.
The little bird is never explicitly given a name, but in one scene Sylvester derisively calls it Cornbread.