It all depends on the certification of the airplane, pilot, and operator and the type of flight conducted.
Airplanes, based on their complexity and such, are certified for a minimum required crew. Some require an SIC, some require a flight engineer, etc.
Pilots on some airplanes can be single pilot qualified if the airplane's certification allows it. An example of this would be the single pilot Cessna Citation business jets. If the pilot is not qualified, then he/she must have a copilot.
The operator in their operations certificate may have restrictions that a single pilot airplane may not actually be flown single pilot even if the pilot and aircraft allow it.
Also, the type of flight conducted makes a difference. In some 135 and smaller 121 operations, a flight may be flown single pilot if it is a cargo flight but it is required to have 2 pilots if it is the exact same flight/aircraft, but it is a passenger flight.
There are other various exceptions out there for all sorts of things. For example, in certain cases, an autopilot may be used as a substitute for a copilot.
All in all, most small general aviation planes are flown single pilot. Most 121 airline flights are flown with at least two pilots. For bigger general aviation planes (ie: biz jets) and 135 (charter, on-demand, ...) it can get really confusing really quickly and is on a case by case basis.