Often used at the end of sentences, when trailing off and not finishing a thought... The three periods have also gained popularity as a feeling on the Internet as well, often being used to express annoyance.

For example:

Person 1: (enthusiastic) Hey, Person 2!
Person 2: (hesitant) Um, hi, Person 1...
Person 1: Wanna come over to my place later and see my collection of used tissues?
Person 2: ...
Person 1: (oblivious) Well, do ya?
Person 2: What do you think?
Person 1: Great! See you at 8!
Person 2: ...

The number of periods used varies from person to person, but usually number three or four.