VFR stands for Visual Flight Rules. Flights conducted under VFR must fly in conditions of visibility that are above certain minimums, so that other aircraft can be seen and avoided by pilots. VFR flight requires that pilots stay a certain distance away from clouds, and requires that the pilots be able to see a certain minimum distance out the windows. Flight inside clouds and in weather foggy enough to push visibility below these minimums is prohibited under VFR.
IFR stands for instrument flight rules. Flights conducted under IFR can be flown even in conditions of little or no visibility, and there are no restrictions on flying through fog or clouds. In most cases, air traffic controllers help aircraft to maintain separation among themselves using radar, and the aircraft also have on-board radar that serves a similar purpose. Flight under IFR can be conducted just by looking at the instruments in the cockpit, without ever looking outside the windows.
Most private pilots fly under VFR most of the time, because it's easier. However, flying VFR does limit a pilot to flying in good weather. VFR flights don't require a flight plan, but a flight plan is still a good idea.
Commercial airline flights fly under IFR, even when the weather is good. IFR is more complex than VFR but it's also much safer, since it doesn't depend on being able to see clearly outside the airplane. Flying IFR requires that a pilot receive extra training. It is much more disciplined and regulated than VFR. The pilots of an airliner flying IFR can fly to their destination and land safely even with cardboard covering the windows. IFR flights must be filed in flight plans.
The abbreviations VFR and IFR refer only to the rules under which a flight is carried out, not to weather conditions. Weather conditions are designated as VMC or IMC. VMC stands for visual meteorological conditions, which means weather conditions in which both VFR and IFR flights are legal. IMC stands for instrument meteorological conditions, which means weather conditions in which only IFR flights may be conducted.