Zoroastrian works two ways. As a noun it means a person who follows the religion founded by Zoroaster: 'her grandfather was a Zoroastrian'. As an adjective it describes anything belonging to that religion: 'a Zoroastrian temple', 'Zoroastrian beliefs', 'the Zoroastrian new year'. Note that the dictionary line above actually describes the religion itself — that religion is Zoroastrianism; this word is the follower and the describing word.

The faith began in ancient Persia, modern Iran, and was once the religion of a great empire. Its followers believe in one supreme good god and hold that good and evil are locked in a long struggle in which humans must take the side of good — good thoughts, good words, good deeds — and that good wins in the end. Fire is kept burning in their temples as a symbol of purity; because of this, Chinese has long called it 拜火教, 'the fire-worshipping religion', though followers themselves will tell you they worship God, not the fire. Today the numbers are small: most Zoroastrians are the Parsees of India, plus communities in Iran, Britain, Canada and elsewhere.

Always a capital letter, like Christian or Buddhist. Formal in tone; you will meet it in history lessons, news about religion, and travel writing.