The symbol itself comes from Aristophanes of Byzantium who invented the system of punctuation where the height of placement of a dot on the line determined its meaning. A dot at the top indicated the end of a thought; a dot in the middle indicated the end of part of a complete thought and a dot at the bottom indicated the end of part of a thought. These are roughly equivalent to the period/full stop, semicolon and comma.
The period of full stop is also used at the end of some abbreviations.
Finally 3 dots indicates a continuation, trailing off of thought, the written equivalent of speech trailing off into nothing:
I wonder if...
Some commentators have noticed that the way in which periods are being used is changing in texting and IMing (Instant Messaging) contexts such as WhatsApp and Twitter. One suggestion is that a period is used to conclude a conversation whilst a line break is used where in the past a period would be used.
Research has found that college students use final period punctuation as little as 29%. In general, it appears that using a period in IMing is not happening. This appears to leave the conversation open for further comments; a final period closes it off.
For more on this idea, see this article. in the New Republic which also suggests that a period is a punctuation mark expressing annoyance.Image © Neal