The title case styles, briefly
APA is the style of the American Psychological Association, used across psychology and much of academia. It capitalizes every word of four letters or more and lowercases shorter articles, conjunctions and prepositions.
AP is the Associated Press style used by most newsrooms. In practice it works like APA for titles: words of four letters or more are capitalized, short function words are not.
Chicago follows The Chicago Manual of Style, the standard for book publishing. It lowercases articles, the conjunctions and, but, for, or and nor, and every preposition no matter how long, so 'through' and 'between' stay lowercase.
MLA is the Modern Language Association style used in the humanities. Like Chicago it lowercases all prepositions, and it also lowercases all seven coordinating conjunctions, including so and yet.
Bluebook is the citation manual for US legal writing. Its rule 8 lowercases articles, conjunctions and prepositions of four or fewer letters, so 'with' stays lowercase but 'between' is capitalized.
AMA is the American Medical Association style used in medical journals. It lowercases articles, coordinating conjunctions and prepositions of three or fewer letters.
NYT follows The New York Times Manual of Style and Usage, which keeps a fixed list of short words lowercase, including 'vs.' in headlines.
Sentence case is not a title style but sits at the end of the section because it answers the same question. It capitalizes only the first word of each sentence and the pronoun I, the way most European publications set their headlines.
Every style capitalizes the first and last word of a title and the first word after a colon. Words with their own capitals, like iPhone or NASA, are left alone unless the whole input arrives in all caps.