"Al-Azhar is a famous Islamic university."
Translation:اَلْأَزْهَر جامِعة إِسْلامِيّة مَشْهورة.
There is no correct order for adjectives in Arabic, but the order modifies slightly the meaning.
- جامِعة إِسْلامِيّة مَشْهورة = It's an Islamic university. Moreover, among all Islamic universities, this one is famous (it doesn't mean that this university is particularly famous in general)
- جامِعة مَشْهورة إِسْلامِيّة = It's a famous university. Moreover, among all famous universities, this one is Islamic
It is like in English: the adjective closest is the most important description, the second closest is just an additional description and so on.
Unlike in English, adjectives follow the noun, so, if you apply the above rule you have to invert the order of the adjectives when translating.