For the vast majority of people, Body Mass Index is a reasonable surrogate for "fatness". However, it does not work as well for very tall people, or highly muscular people. Nor does it take into account the general tendency for BMI to increase with age, or the differences in BMI between men and women.
It's popular mainly because it is easy to calculate, and because it is a meaningful number for most people (i.e., most folks with a BMI of 30 really are "obese").
www.halls.md is one of the best sites on the web for understanding BMI. The site also has some very interesting comparisons based on age and gender, and in-depth analysis of the limitations of BMI.