Definition Attributable risk is the excess risk that is associated with your exposure of interest.
Explanation
Let's face it, bad things happen all the time for lots of different reasons. The attributable risk is a way of quantifying how many of the bad things are due to a particular exposure versus how many would have happened anyway.
Example
Risk of developing lung cancer
for nonsmokers: 0.0001
for current smokers: 0.001 
Attributable Risk = 0.001  0.0001 = 0.0009 

In this example, nonsmokers are expected to develop lung cancer at a rate of 10 cases per 100,000 population per year. Smokers on the other hand, have a lung cancer rate of 100 per 100,000 per year. This is an excess risk or attributable risk of 90 per 100,000 per year. (Assuming of course, that the exposure is causal.)
Expansion
If a large proportion of your outcome is due to your exposure of interest (e.g. smoking and cancer), certain study designs will be more appropriate than if a small proportion of your outcome is due to your exposure (e.g. anthrax and cough).
From a public health perspective, there are several associated measures of interest:
Attributable risk percent (fraction) for the exposed 
X% of outcomes among the exposed are attributable to the exposure 
Attributable risk percent (fraction) for the population 
X% of outcomes in the population are attributable to the exposure 

Risk of developing lung cancer
for nonsmokers: 0.0001
for current smokers: 0.001

Risk Ratio = 10
Attributable Risk = 0.0009
Attributable Fraction for exposed = 90%

Prevalence of smoking in the population = 0.33 
Attributable Fraction for the population = 75% 
 
A cynic/stooge for the tobacco industry might look at a 10fold increase in risk of developing cancer for smokers and scoff, because 10 x 0.0001 is still quite small. However, the attributable fraction for the exposed shows us that among smokers who develop lung cancer, a whopping 90% of these cases can be attributed to their smoking. Furthermore, the attributable fraction for the population shows us that 75% of lung cancers are due to smoking (this is very high). Considering that well over 100,000 cases of lung cancer are diagnosed in the U.S. each year, it is easy to see why eliminating smoking is a public health priority.
It is important to note that the attributable fraction for the population is dependent upon the prevalence of the risk factor in the population. As the prevalence of exposure declines, the attributable fraction declines as well. Using the above numbers, but with a 15% prevalence of smoking, the attributable fraction for the population drops to 57%.
Further reading
Analytical methods used by epidemiologists
*Formulas from Kelsey et. al, Methods in Observational Epidemiology, pp3740.
