The ellipsis, better known as "the three dots thing" or "dot-dot-dot," 
stands in for omitted words in quoted material. Wikipedia has a good overview, although it includes a bunch of other uses, such as in computer programming, mathematical notation, and foreign languages.
As others in this thread have pointed out, there are a few variations in style, such as spaces between the dots or even use of asterisks instead of dots, which vary according to which stylebook you're using or the industry for which you're writing.
Sometimes writers incorrectly use the ellipsis to indicate a pause (where they should be using a dash, or possibly a comma). The other common transgression is use of the terminal ellipsis to indicate a "trailing off" at the end of a sentence rather than the omission of quoted material.
Sadly, I no longer have the AP or U.S. News & World Report style guides I used to own -- but a quick check of several writing references (On Writing Well by William Zinsser, The Careful Writer by Theodore Bernstein, The Transitive Vampire by Karen Elizabeth Gordon, Strunk & White, et al.) shows that the ellipsis doesn't preoccupy many authorities on writing. It's more in the domain of the Chicago Manual of Style, I suppose, another book I no longer own.
In my years as an office worker, where I became known (notorious) as a grammar and usage expert (fussbudget), the ellipsis was one of the things people most often asked me about, probably to settle an argument with their boss. The other was how to punctuate bulleted lists. But let's not go into that!
Edit history
= new reply since forum marked as read
