Persian points at a place and at everything that comes from it. The country is called Iran today; its older name was Persia, and this adjective kept the older name. So you meet Persian mostly in front of a noun: a Persian carpet, Persian food, the Persian Empire, a Persian cat. It can point at the people of that country, at the language they speak, or at things made there or made in that style. It always keeps its capital P, because it is built on the name of a place.

This is the noun for a person, so it takes a and it has a plural: a Persian, the Persians. You meet it most often in writing about history — the Persians of the old empires. For someone living in Iran now, English usually says an Iranian, and many speakers hear a Persian as a dated way to say it. People who do call themselves Persian today usually mean the ethnic group whose first language is Persian, which is not the same as everyone who lives in Iran.

This is Persian as the name of a language, so it behaves like English, Japanese or Spanish in a sentence. It takes no a and no plural: she speaks Persian, he is learning Persian. To say what something is written or sung in, use in: the sign was in Persian. The same word can turn back into an adjective in front of a noun, as in Persian poetry, and there the meaning is still the language.