For example, if you own a home worth $300,000 and sell it to a family member for $200,000, they've received a gift of equity of $100,000. A gift of equity can occur if a home is given away for no compensation or if a discount is offered on its value.
Yes, if you have a mortgage on the property your lender will need to consent to any other parties becoming legal owners. They will require the incoming party to become a party to the mortgage as well.