In The Vampire Diaries Katharine is pregnant. Who is the father of Katharine's child, why did she lose the baby, and why was her father upset with her??
The father is unknown from the story.
As to why Katherine had to give the baby away, you have to understand in the 1500's it was "improper" and "embarrassing" for the family to have a young unmarried daughter get pregnant. To hide the family's shame, Kathrine's father gave the child away to an anonymous family. Because of the shame Kathrine's actions would bring to the family, her father was extremely angry at her.
Katherine begged her father to allow her to hold the child just once, but he said that she had disgraced the family, and took the child away to a new family, so that the shame would not become known. - wiki
I verified personally in the episode the material of this quote from the wiki.