A lean running condition means that the air/fuel mixture is unbalanced, and that the engine is getting either too much air, not enough fuel, or a combination of both. Optimally, this ratio should be around 14.7 parts air to 1 part fuel by mass, at low or cruising loads. This ratio is known as stoichiometric for gasoline. Ideally, at WOT, you should aim between 11.5:1 (Rich best torque at WOT) and 13.3:1 (Lean best torque at WOT).
Running lean, like having your spark timing too advanced, creates much more heat during the combustion process. This can lead to piston seizure, a hole in the piston, melted spark plug electrode or other complications due to overheating. You will not be producing peak power, and your exhaust will also emit more oxides of nitrogen, arguably one of the worst byproducts of gasoline combustion for our environment.
Preforming a plug chop is a good way to find out if your engine is running too lean, as you will see a powdery white residue on your spark plug in the presence of a lean condition.
On the other end of the spectrum, richer ratios will also cause lost power, however this will most likely be accompanied by black smoke (around the 9:1 fuel/air ratio area), and fouled spark plugs.