The short answer is Anticollision Lights.
The long answer:
The regulation that specifies the required instruments and equipment is 91.205. So the answer may be found at faa.gov->regulations and policies->faa regulations->current federal aviation regulations. I've quoted it below, but no lights are required for an aircraft with a Standard Airworthiness Certificate (Normal, Utility, Acrobatic, Commuter, Transport)for Day VFR except that an aircraft manufactured under FAR Part 23 after March 11, 1996, must have an approved aviation red (I'm pretty certain that this will always be a rotating beacon) or aviation white anticollision light system (I'm pretty certain that this will always be a strobe light system).
Part 23 is the regulation that governs the manufacture of aircraft with in the Normal, Utility, Acrobatic, and Commuter Category. No ultralights, no jets, basically small aircraft.
The longest answer:
The regulation that specifies that the lights need to be on is 91.209
No person may
b) Operate an aircraft that is equipped with an anticollision light system, unless it has lighted anticollision lights. However, the anticollision lights need not be lighted when the pilot-in-command determines that, because of operating conditions, it would be in the interest of safety to turn the lights off.
OK, SO THE ABOVE REG SAYS THAT IF YOU HAVE THEM YOU MUST USE THEM EXCEPT...
(a) General. Except as provided in paragraphs (c)(3) and (e) of this section, no person may operate a powered civil aircraft with a standard category U.S. airworthiness certificate in any operation described in paragraphs (b) through (f) of this section unless that aircraft contains the instruments and equipment specified in those paragraphs (or FAA-approved equivalents) for that type of operation, and those instruments and items of equipment are in operable condition.
(b) Visual-flight rules (day). For VFR flight during the day, the following instruments and equipment are required:
(1) Airspeed indicator.
(3) Magnetic direction indicator.
(4) Tachometer for each engine.
(5) Oil pressure gauge for each engine using pressure system.
(6) Temperature gauge for each liquid-cooled engine.
(7) Oil temperature gauge for each air-cooled engine.
(8) Manifold pressure gauge for each altitude engine.
(9) Fuel gauge indicating the quantity of fuel in each tank.
(10) Landing gear position indicator, if the aircraft has a retractable landing gear.
(11) For small civil airplanes certificated after March 11, 1996, in accordance with part 23 of this chapter, an approved aviation red or aviation white anticollision light system. In the event of failure of any light of the anticollision light system, operation of the aircraft may continue to a location where repairs or replacement can be made.
(12) If the aircraft is operated for hire over water and beyond power-off gliding distance from shore, approved flotation gear readily available to each occupant and, unless the aircraft is operating under part 121 of this subchapter, at least one pyrotechnic signaling device. As used in this section, “shore” means that area of the land adjacent to the water which is above the high water mark and excludes land areas which are intermittently under water.
(13) An approved safety belt with an approved metal-to-metal latching device for each occupant 2 years of age or older.
(14) For small civil airplanes manufactured after July 18, 1978, an approved shoulder harness for each front seat. The shoulder harness must be designed to protect the occupant from serious head injury when the occupant experiences the ultimate inertia forces specified in §23.561(b)(2) of this chapter. Each shoulder harness installed at a flight crewmember station must permit the crewmember, when seated and with the safety belt and shoulder harness fastened, to perform all functions necessary for flight operations. For purposes of this paragraph—
(i) The date of manufacture of an airplane is the date the inspection acceptance records reflect that the airplane is complete and meets the FAA-approved type design data; and
(ii) A front seat is a seat located at a flight crewmember station or any seat located alongside such a seat.
(15) An emergency locator transmitter, if required by §91.207.
(16) For normal, utility, and acrobatic category airplanes with a seating configuration, excluding pilot seats, of 9 or less, manufactured after December 12, 1986, a shoulder harness for—
(i) Each front seat that meets the requirements of §23.785 (g) and (h) of this chapter in effect on December 12, 1985;
(ii) Each additional seat that meets the requirements of §23.785(g) of this chapter in effect on December 12, 1985.
(17) For rotorcraft manufactured after September 16, 1992, a shoulder harness for each seat that meets the requirements of §27.2 or §29.2 of this chapter in effect on September 16, 1991.
THE ABOVE REG SAYS THAT ARE REQUIRED IF THE AIRCRAFT WAS MANUFACTURED AFTER 3/11/96 THEY MAY BE INOP BUT ONLY TO BE FLOWN TO A PLACE WHERE REPAIRS MAY BE MADE.
Notice the reference to Standard Airworthiness Certificate which is defined in FAR 21.175 to be
"Standard airworthiness certificates are airworthiness certificates issued for aircraft type certificated in the normal, utility, acrobatic, commuter, or transport category, and for manned free balloons, and for aircraft designated by the Administrator as special classes of aircraft."
(b) Special airworthiness certificates are primary, restricted, limited, light-sport, and provisional airworthiness certificates, special flight permits, and experimental certificates.
THE ABOVE SAYS THAT ALL THIS APPLIES TO REGULAR AIRCRAFT, NOT ULTRALIGHTS, ETC.