In 1 Chronicles 3:5 she is called Bath-shua.
Bathsheba means the seventh daughter or the daughter of the oath.
2 Samuel 11:1 to 12:25 tells the story of David's adultery with Bathsheba, and his subsequent murder of Uriah in order to conceal his guilt. His plan comes unstuck when God sends the prophet Nathan to denounce David by means of a parable. David is completely taken in, declaring at the end of it "The man who did this deserves to die!" only to be told by Nathan "You are that man".
Some find it surprising that such a sordid affair could be part of the ancestry of Jesus Christ, and more so that there is no attempt to hide it. In fact David is far from unique among Jesus' ancestors in being seriously flawed, see Jacob.