The term Arab refers to a person of Arab descent. As an adjective, it describes something associated with Arabs. Meanwhile, Arabic is used only for the language, and Arabian refers to objects or concepts (not people) related to the Arabian Peninsula (e.g., Arabian Gulf).
In Arabic, the word عَرَبِيّ is commonly used in all these contexts. However, when referring to Arabian, the adjective خَلِيجِيّ (Gulf) is often used, such as in الخَلِيج العَرَبِيّ (the Arabian Gulf), and not الخَلِيج الفَارِسِيّ (Persian Gulf), which should be avoided. For instance:
• “Arab history is very rich in powerful rulers.”
إِنَّ التَّارِيخَ العَرَبِيَّ حَافِلٌ بِأَسْمَاءِ قَادَةٍ (حُكَّامٍ) أَقْوِيَاءَ.
• “I bought an Arabic book on Islamic Spain.”
اِشْتَرَيْتُ كِتَابًا بِاللُّغَةِ العَرَبِيَّةِ عَنْ الأَنْدَلُسِ (إِسْبَانِيَا الإِسْلَامِيَّةِ).
• “Arabian horses are famous all over the world for their speed and grace.”
تَشْتَهِرُ الخُيُولُ العَرَبِيَّةُ فِي العَالَمِ بِسُرْعَتِهَا وَرَشَاقَتِهَا (حُسْنِهَا وَرَشَاقَتِهَا).
• “The remaining part will be exported to Gulf markets, especially to Oman and Kuwait.”
سَيَتِمُّ تَصْدِيرُ الجُزْءِ البَاقِي إِلَى الأَسْوَاقِ الخَلِيجِيَّةِ، خَاصَّةً إِلَى عُمَانَ وَالكُوَيْتِ.
When discussing architecture, the term عَرَبِيّ is often used in Arabic. However, in English, it is more appropriate to translate it as Islamic for idiomatic reasons. For example:
• “Islamic architecture spread in the Middle Ages.”
اِنتَشَرَتْ هَندَسَةُ المِعْمَارِ العَرَبِيَّةِ/العِمَارَةُ الإِسْلَامِيَّةُ فِي القُرُونِ الوُسْطَى.
Finally, a phrase like هُوَ خَلِيجِيّ is best translated as “He comes from the (Arabian) Gulf.”