The Wikipedia article: Colon_(punctuation) gives (reformatted; I've italicised the most relevant comments) a good overview of the uses and conditions of use of the colon:
The most common use of the colon is to inform the reader that what
follows the colon proves, explains, defines, describes, or lists
elements of what preceded it.
In modern American English usage, a complete sentence precedes a colon, while a list, description, explanation, or definition follows
it. The elements which follow the colon may or may not be a complete
sentence: since the colon is preceded by a sentence, it is [conventionally regarded as] a complete
sentence whether what follows the colon is another sentence or not.
While it is acceptable to capitalize the first letter after the colon
in American English, [this] is not [usually considered to be] the case in British English.
colon used before list
Williams was so hungry he ate everything in the house: chips, cold
pizza, pretzels and dip, hot dogs, peanut butter and candy.
colon used before a description
Jane is so desperate that she'll date anyone, even Tom: he's uglier
than a squashed toad on the highway, and that's on his good days.
colon before definition
For years while I was reading Shakespeare's Othello and criticism on
it, I had to constantly look up the word "egregious" since the villain
uses that word: outstandingly bad or shocking.
colon before explanation
I had a rough weekend: I had chest pain and spent all Saturday and
Sunday in the emergency room.
Some writers use fragments (incomplete sentences) before a colon for emphasis or stylistic preferences (to show a character's voice in
literature), as in this example:
Dinner: chips and juice. What a well-rounded diet I have.
I'm sure that all the (four) listed types of use here have been used in at least less formal writing, for dramatic effect or conciseness.
The colon is also used in what The Punctuation Guide (J.R.'s link above) calls 'non-grammatical' ways (in registers where insistance on formal grammar would be silly), which include:
The colon is frequently used in business and personal correspondence.
Dear Ms. Smith:
cc: Tom Smith
Attention: Accounts Payable
Don’t forget your swimsuit.
But 'non-grammatical' doesn't mean ungrammatical / unacceptable.