For language = Arabic "I speak Arabic" For people and things = Arabic or Arab. "She is Arab" or "She is Arabic" "I like Arab food" or "I like Arabic food" For animals and places = Arabian "The Arabian peninsula" or "Arabian birds" or the Arabian nights!

Arabic is the language. Arab is the culture and ethnicity. Arabian is an old-fashioned term, nowadays mainly used in certain names, such as the Arabian peninsula. A writer might chose to use this word for romantic effect (eg to name a perfume!) .

Honestly, this is something that even native English speakers would always confuse with, as many, in fact almost all native English speakers are actually unable to differentiate between any of them, so you shouldn't be surprise if an American for instance uses the word "Arabic" to refer to to your own ethnicity, however to answer your question, the word" Arab" is usually the one that referrers to one's ethnicity.