Body Mass Index, a measure of human skinnyness or obesity. It is a way to measure if you are overweight or underweight.

Underweight: BMI of less than 18.5
Normal 18.5 - 24.9
Overweight 25.0 - 29.9
Obesity 30.0 - 39.9
Extreme Obesity 40.0 and greater

Just where to draw the line between normal and overweight is a judgement call. There are slight increases in risks of diabetes and heart disease with BMIs over 22. These risks become dramatically larger with BMIs over 30.

As noted over at ideal body weight, the following formula is an estimate that works for 90% of adults, but not for athletes, the aged or adolescents under 20 years of age. f you carry a lot of muscle then BMI won't work for you. If you think that you fall outside this formula's competence, then use a more accurate measurement technique to ascertain how much body fat you carry.

Formula: BMI = w / (h * h), where w is weight in kilograms, and h is height in meters.
Or a version with the conversion from imperial units built in:
BMI = w * 704.5 / (h * h), where w is weight in pounds and h is height in inches.

Worked example for my BMI:
I am 1.80m tall (just under six foot)
I weigh 72kg (147 pounds)
My BMI = 72 / (1.8 * 1.8) = 22.22

So now that I know that I am within the ideal weight range, how big is my range? Rearranging the equation for weight gives:
w = BMI * h * h
The ends of the ideal weight range correspond to a BMI of 18.5 and 25.0. Given my height, this gives
Low: w = 18.5 * 1.8 * 1.8 = 59.94 kg (131.9 pounds)
High: w = 25 * 1.8 * 1.8 = 81 kg (178 pounds)
Note that this calculation seems to be quite sensitive to small changes in height (using a height of 1.805 m instead yields a minimum weight of 60.27kg) so try to get that accurate.
Given that the input numbers of height and weight are only accurate to two digits, the output probably has the same degree of accuracy. Thus my ideal weight range is 60 to 81 kg.

Sources: Sites that Google turned up and the source code of their javascript BMI calculators. All calculations were tested in Excel