Complex airplane is a term used by the U.S. Federal Aviation Administration to describe a light propeller-driven aircraft with a particular set of characteristics. In order to act as pilot in command of an airplane which has these characteristics, a pilot must meet several requirements. If an aircraft is above 12,500 lbs takeoff weight, or is powered by turbojets or turbofans, the pilot will require a type rating specific to that aircraft.
First of all, what is a complex airplane? Well, according to 14 CFR § 61.1(b)(3) (Title 14, Code of Federal Regulations - Aeronautics and Space):
Complex airplane means an airplane that has a retractable landing gear, flaps, and a controllable pitch propeller, including airplanes equipped with an engine control system consisting of a digital computer and associated accessories for controlling the engine and propeller, such as a full authority digital engine control; or, in the case of a seaplane, flaps and a controllable pitch propeller, including seaplanes equipped with an engine control system consisting of a digital computer and associated accessories for controlling the engine and propeller, such as a full authority digital engine control.
In other words (my own, so trust the ones above rather than mine if you're confused) a complex airplane is one which has three things: retractable landing gear, flaps, and a variable pitch (controllable pitch) propeller. These are all systems which require pilot management during flight, and which some simpler airplanes do not have. They do, as the name implies, make the airplane more complex. As a result, pilots must meet certain requirements to be allowed to operate them, to be sure the pilot has adequate training and experience (and won't, you know, leave the landing gear up when they hit the runway - this is why we have GUMPS). Even if there are computers managing those settings for the pilot, the pilot must still be appropriately trained because hey, computers can break or even do dumb and dangerous things by design - and the pilot is the pilot in command and legally responsible for everything about the flight of that aircraft, including what the computers decide to do.
What are those requirements? Let's turn back to 14 CFR, this time § 61.31(e)(1), which states:
Except as provided in paragraph (e)(2) of this section, no person may act as pilot in command of a complex airplane, unless the person has:
(i) Received and logged ground and flight training from an authorized instructor in a complex airplane, or in a flight simulator or flight training device that is representative of a complex airplane, and has been found proficient in the operation and systems of the airplane; and
(ii) Received a one-time endorsement in the pilot's logbook from an authorized instructor who certifies the person is proficient to operate a complex airplane.
There are waivers in place so that if you have acted as a pilot in command of a complex airplane prior to August 4, 1997 you're 'grandfathered' and don't need this endorsement.