Mr. Sweeney is or was the father of Emily Sweeney.
He is mentioned in "The Commitment Determination" when Raj tells Howard and Bernadette that Emily wants to have sex in a graveyard and on somebody's grave and Howard asks "Like, a random person or somebody she knew?" and Raj asks what difference does that make and Howard replies "Well, if it’s her father’s grave and they didn’t get along, then you know she holds a grudge.".
Because of what Howard said, it is a hint that Mr. Sweeney is dead, but this has been unconfirmed.