Ellipsis (plural ellipses) is a punctuation mark or series of marks that usually show something is missing.
For example, ellipsis:
The missing words or thoughts need to be supplied by the listener or reader and are thus often open to different interpretations.
In writing, we usually use three dots to indicate an omission. This can be used to leave out information not strictly necessary for the context. It's often done when quoting from a book or other source.
For example, the first sentence here is in full. The second sentence quotes from the first but leaves out non-essential information that isn't needed there and then:
Allied forces, desperate for a victory after recent setbacks, managed to secure the beachhead successfully.
Allied forces ... managed to secure the beachhead successfully.
Some style guides use the three dots inside square brackets to indicate that a piece of the original has been purposefully removed.
Allied forces [...] managed to secure the beachhead successfully.
Note that whenever three dots are used to show removed text they should have a space before and after.
Three dots can also be used to indicate a meaningful pause:
Dracula turned to me and spoke, "I never drink... wine."
Here the implication is that he drinks something else.
Note that the three dots here follow on immediately from the previous word, i.e. there is no space before they begin.
Finally, in writing ellipsis indicates a trailing off of thought or idea:
She turned to me and gently took my hand. "Why don't we..."
Depending on the context, the reader can imagine the missing words.
Note, again there is no space before the three dots begin.