What kind of bond occurs when both of the electrons in a molecular bond originate from the same atom?
A bond which is covalent but where both electrons originate from the same atom can often be found where one species has lone pairs to donate, for example nitrogen based compounds such as ammonia.
In such cases the lone pair may interact with incomplete orbitals on another atom and form a bond whereby both electrons come from the lone pair, and none from the atom to which they bond. This is the way that such things as complex ions are formed.
This type of bonding is known as coordinate or dipolar bonding (occasionally "dative covalent").