This is a good question, because there’s no distinction in Arabic between these three words.

The people are Arabs. The culture is Arab. The cuisine is Arab.

The language is Arabic. “Arabic” is also used for a few other things, like Arabic coffee (not to be confused with arabica coffee. Arabic coffee is a special way of brewing arabica coffee popular among Arabs) and Arabic numerals.

The word “Arabian” is quite outdated. It should be noted that it actually only refers to things from the Arabian Peninsula, not the Arab world at large. However, it’s not really used much anymore. The main phrases with this word are “Arabian horses” and “Arabian Nights” (the book). It’s also used in the phrase “Saudi Arabian” to refer to someone from Saudi Arabia. I would not refer to people from the Arabian Peninsula at large as Arabians. They’re Arabs, just like the people in the rest of the Arab world.

It’s interesting to note that there’s actually a language called Modern South Arabian, and it’s in fact a completely different from Arabic, but it’s spoken in South Arabia (mostly Yemen), hence why it has that name.

Arab: person (male or female), usually used with 'an', ie 'She is an arab'

Arabic: language only - 'He speaks Arabic', 'It is written in Arabic' or 'It is written in the Arabic alphabet'

Arabian: something from the Middle East, roughly. Arabian food, Arabian horse.

Hope that helps!