Benford’s Law says that if you take a list of numbers from certain processes, lots of them will start with “1″ and very few of them will start with a “9″. This is because we use a base-10 counting system, so if something is growing at a steady average rate it takes just as long to get to 2000 as it took to get to 1000. eg: 139, 253, 443, 463, 585, 745, 884, 1028, 1108, 1299, 1424, 1514, 1531, 1710, 1818, 2051*. The chances that you sample it when it starts with a “1″ are higher than the chance that you sample it when it starts with anything else.
Benford’s Law is good at detecting financial fraud because financial calculations have patterns that cause steady average grow: most importantly, multiplying quantities and prices. Since most electoral systems divide voters into equal-sized voting areas, the votes in each area don’t grow at a steady rate. So instead electoral analysts skip the first digit and apply Benford’s Law to the later digits in results.
This analysis assumes that votes for each candidate will grow at a steady rate. In other words, if you combine two voting areas, the results for each candidate should, on average, double. But representational democracies allocate representatives based on the regions precisely because that assumption is wrong.
Other election analysis assume that voters, when averaged together, act completely randomly. If that assumption is correct, I’m not sure why fraudulent elections are a bad thing. 
Statistical analysis to detect election fraud is a very new field. Much of the work is being done in the US, which means researchers have certain biases:
- they don’t understand the boundaries of the places they’re analyzing
- they have incentives to find fraud in the elections of America’s enemies
- they have disincentives to find fraud in the elections of Western democracies (in particular it is taboo to analyze the 2000 Presidential election)
- they assume divisive bipartisan politics
I’d love to see a paper where someone analyzes a real election known to be fraudulent rather than a simulation or at least applies a method to a large, broad sample of elections.
* I generated these using a uniform distribution from 0 to 200.