VFR means visual flight rules. A VFR pilot navigates and flies by looking out the window. He uses the view out the window to keep the aircraft straight and level, and he navigates from place to place by looking at things on the ground (roads, rivers, buildings, etc.). He can also use instruments if he wants, but the law requires that he be able to see and fly visually (flying in clouds or when the ground isn't visible is not allowed, and in some countries flying at night isn't allowed, either). VFR pilots also must keep their eyes open for other airplanes nearby so that they don't hit anyone.
IFR means instrument flight rules. An IFR pilot navigates using instruments in the cockpit. It is not necessary for him to look out the window, and in fact a pilot flying IFR can fly to his destination even if the windows are covered by cardboard. Everything is done by instruments. Since it's often not possible to see outside when flying IFR, IFR flights work in conjunction with air traffic controllers, who use radar to advise IFR flights of other aircraft in the area, thereby maintaining a safe distance between them.
IFR is safer than VFR, because it can be carried out in any type of weather, regardless of visibility. However, flying IFR is much more complicated than VFR, and requires much more training and practice.
Airline pilots flying for commercial airlines are required to fly IFR, for safety reasons. Even in perfectly clear weather, the flight is still conducted under IFR. Of course, the pilots do look out the windows when flying IFR, but the important thing is that they can fly safely even if nothing is visible outside the windows.
When airline pilots are flying on their own for fun, in a small plane for example, they can fly VFR if they want. But when they are on the job and flying an airliner, IFR is required.