[noun] + 之 + [noun]
[noun] + 之 + [adjective]
[verb] + 之
之 is the written counterpart of 的 between a modifier and a noun. It belongs to formal writing, sayings and set phrases; in conversation the same idea takes 的.
之 sits best between short words, often a single syllable on one side: 无价之宝 "a priceless treasure", 一家之主 "the head of the family", 一日之功 "one day's work". With a long modifier, a clause or a pronoun, 的 is the normal choice.
Every fraction is built on 之: 三分之一 "one third", literally "of three parts, one". 百分之十 is "ten percent". Fractions and percentages have their own entry.
In writing, 之 can turn "the city changed a lot" into "how much the city changed", which then serves as the topic of a comment. It's a way of saying "so … that".
In set phrases 之 is the object of the verb before it: 置之不理 "leave it and ignore it", 随之 "along with it", 为之 "for it".
Words such as 之一, 之间, 之前, 总之 and 之所以 contain 之 but are learned whole, and have their own entries.