What are the correct ways to use a colon?

between a grammatically complete introductory clause (one that could stand as a sentence) and a final phrase or clause that illustrates, extends, or amplifies the preceding thought. If the clause following the colon is a complete sentence, it begins with a capital letter.

• Freud (1930/1961) wrote of two urges: an urge toward union with others and an egoistic urge toward happiness.
• They have agreed on the outcome: Informed participants perform better than do uninformed participants.

in ratios and proportions

• The proportion (salt:water) was 1:8.

in references between place of publication and publisher

• New York, NY: Wiley. St. Louis, MO: Mosby.

after an introduction that is not an independent clause or complete sentence.

• The formula is r_i = a_i + e.

(adapted from the sixth edition of the APA Publication Manual, © 2010)