These two verbs sound the same (in-DIGHT) and have a common origin, but their meanings are very different now—and only indict is commonly used.
Indite is archaic and almost never used. It means “to write down” or “compose a piece of writing”—but most English speakers do not recognize this word (and they will think it is indict if you say it).
Indict is a common and useful word concerning the justice system (laws and courts); it means “to formally find someone guilty of breaking the law, by jury”; one indicts someone on charges (e.g. the charge of murder).
The jury indicted him on all charges, and he went to jail.
Some English dictionaries say that indict means “accuse or charge”; that is, “to say that someone is guilty of a crime”; but in law, indict only means “to officially find someone guilty”:
Although many believed he was guilty, they could not indict him.
However, indict can also be used metaphorically, usually in its noun form, indictment, to say that someone or something is just wrong or bad:
Many people view the economic inequality of many Americans is an indictment of capitalism.
The legal event when someone is indicted in court, is also called an indictment. You will hear these words on the news, quite often.