Arabian means of Arabia — the peninsula — or of its people in an older, place-tied way. The set phrases are the ones to learn: the Arabian Peninsula, the Arabian Desert, the Arabian Sea, the Arabian Nights (the old story collection), and the Arabian horse, a famous slim, fast breed. As a noun, an Arabian can mean that horse, or, less often now, a person from Arabia. Though actually, for people today we nearly always say Arab, not Arabian. Arabian feels a bit storybook or geographic: good for land, sea, horses and old tales; a little odd for a living neighbour. Capital A. It is not the name of the language (that is Arabic).