The answer to this question depends on the situation. We can roughly classify all electrical systems into two categories: static electricity systems and circuit electricity systems. Note that all electrical effects are actually part of one unified set of physical laws. This classification is therefore ultimately arbitrary and over-simplified. However, this classification is sufficient for our current purpose of understanding electric current.

Public Domain Image, source: NOAA.A static electricity system involves the flow of electric current as a result of a buildup of electric charge somewhere. Such a system does not involve a closed electrical circuit. Examples of this type of system include lightning and the sparks you get when you rub your feet on a carpet. Electrons naturally repel each other. When a lot of electrons get piled up in one place, they can push on each other so strongly that some of the electrons get pushed right off of the object. They end up getting pushed out through the air, the water, or whatever surrounds the object. We call a collection of moving electrons an electric current, therefore a buildup of charge can drive a current. The electrons simply flow away from the pile and ultimately end up attached to atoms in the environment. In this way, we can have an electric current even if we don't have a complete electrical circuit. In air, an electrical current takes the form of dark discharge, corona discharge, or sparks (depending on if the current is weak, medium strength, or strong, respectively). Note that the name "static electricity" is a poor name since the electric charge is not always stationary in this type of system. More accurate names would be "non-circuit electricity" or "charge buildup electricity."