It depends — "on" is lowercase in AP and Chicago Style but capitalized in NYT Style.
"On" splits directly along the line between journalism and publishing. AP Style lowercases it as a preposition, while NYT Style capitalizes it. Chicago Style and MLA Style lowercase it based on its two-letter length.
NYT Style takes a different view, capitalizing most prepositions including "on." This produces headlines that are visually denser and more uniformly capitalized than AP-style headlines.
Like other prepositions, "on" can also function as an adverb or particle in phrasal verbs — "carry on," "move on," "hold on." When it carries adverbial meaning as part of a verb phrase, Chicago Style recommends capitalizing it.
"On" as the first or last word of a title is always capitalized. Titles beginning with "On" — a common form for essays and philosophical works — are correctly capitalized regardless of what style guide applies to the rest of the title.
"On" is among the shortest prepositions in English and is lowercased in AP Style, Chicago Style, MLA Style, and AMA Style. These style guides treat all short prepositions as function words that do not carry enough independent meaning to warrant capitalization.