I have often heard people saying that the light in the picture X is appearing "flat". I want to understand what exactly does that mean, and what are the possible ways to counter that?
Flat lighting is when a scene, regardless of its type, is largely and broadly "diffusely and directly lit". Flat lighting illuminates the scene, however it does not bring out depth and detail, or add any intriguing character. In some rarer/specific cases, you may require flat lighting, depending on the exact goals you have. In general, however, flat lighting makes a photo look rather dull and lifeless.
With nature photography, flat lighting is often caused by overcast skies, direct sunlight, etc. Landscape photos in particular generally look best when the sun is at a low, indirect angle such that the shape of mountains cast shadows upon themselves, bringing to light that shape and providing depth. Early morning sunrise or evening sunset light is good for this reason, as well as for the fact that such light is often more colorful and vibrant than midday sunlight.
With portrait photography, flat lighting dulls or eliminates the intriguing features of your subjects that give them their character, expression, emotion. Highly diffused, direct lighting can be a useful tool for portraits, however uses as the only source of lighting it can produce rather boring portraits. More complex, angled lighting can do wonders for portrait photography. Use of sunlight or artificial lighting, angling your subjects to light sources, providing adequate and appropriate fill lighting, etc. can greatly enhance the character and emotion of your subjects. I would search our forum for other questions of portrait and studio lighting...there are many skilled photographers here who have provided superb answers detailing specific kinds of lighting setups for portrait photography that go into far greater depth than I ever could here.
It's all about the technique. Overcast flat lighting doesn't neccessarily make for dull lifeless photos. The key to great photos is to see a photo as you take it. Not about the size of your lens guys, which dare I say probably compensates for an inadequacy elsewhere ;-) There is no right or wrong, learn the rules then forget them!
Flat lighting is typically what you see at midday with generous amount of light from the sun. The other end of the spectrum would be what you find in the minutes before and after sunrise/sunset.
Photographers, especially nature photographers will stay away from shooting during midday when the light is flat, because it is unattractive for most purposes. The soft light around the ends of the day is much more interesting, colorful, and pleasant on the eye.
Flat lighting can be achieved in a studio setting by lowering contrast and or shadows through the use of lighting and diffusers. If you are shooting macro shots or close up images, flat lighting can be desirable. But for portrait work, under most circumstances it is not typically used. It can be used for certain effects but I would consider it rarely to be used.
Somewhat Flat Lighting:
Dynamic Lighting with depth and contrast: