Hi sham, in short answer: Arabic and Arabian are adjectives and Arab is a noun,
- Arabic is the language of Arabs or related to the Arab or Arabic
- Arabian: a native or inhabitant of Arabia, related to the nationality o the place s/he came from,:
- he's Arabian, he's from Yemen
- he's Arabic, he's from a country in which they speak Arabic
- Arab here is a noun, means an inhabitant of the Arabia, or a horse originated from this area