VFR stands for Visual Flight Rules. This is the default rule set for private pilots in daylight flight. Navigational instruments are not used as the primary basis of direction. Visual cues from the ground are used to find ones way through the sky. During VFR pilots are required to be aware and look around for other aircraft.
IFR stands for Instrument Flight Rules. Instrument flight is more complicated and involves an additional rating that private pilots can train to receive. For general aviation, IFR is used at night or in unfavorable weather conditions, not normally during the day in clear conditions unless you are at higher altitudes. IFR uses beacons on the ground, usually abbreviated as VOR, which each have their own frequency. The FAA will issue a series of VOR's to fly between and assigned altitudes to fly at. After dialing in the frequency of the VOR in the nav input, the flight instruments will provide all of the necessary directional information. In a nutshell, flying in IFR involves filing a definite flight plan with the FAA and using only the instruments and communication/navigation radios to fly.
All commercial flights operate under IFR, meaning, the pilots on an airliner will be navigating a flight plan and do not have to use visual cues during flight. There are exceptions though. Commercial pilots often perform a visual approach to the runway without use of instrument landing systems. (a part of the IFR) Commercial aircraft follow IFR navigation from takeoff to landing with constant communication with ATC and ARTCC.