What is the difference between pride and arrogance? In which case is pride permitted? Please, kindly give examples of behaviour which indicates arrogance. Is looking down upon people considered arrogance?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger. 

The Prophet defined arrogance (Arabic: Kibr) in his saying, "Arrogance is rejecting the truth and disdaining people." [Muslim]

An-Nawawi, may Allah have mercy on him, said, "As for rejecting the truth, it means denying it and repelling it out of deeming oneself far above it; and ‘disdaining people’ means despising them."

As for pride (Arabic: ‘Ujb), Al-Ghazaali, may Allah have mercy on him, defined it as follows in Ihayaa’ ‘Uloom Ad-Deen, "It is to praise and show off the blessings that one has and to take them for granted while forgetting to ascribe them to Allah, the Bestower of all Blessings…"

Pride, in this sense, is all blameworthy, and none of it is permissible, because forgetting to ascribe the blessings to the Benefactor and taking them for granted and admiring oneself without ascribing the favor to Allah are among the most evil deeds. Pride is one of the reasons for destruction, as the Prophet said, "...and there are three destructives sins: … followed desires, obeyed miserliness, and that a person admires himself (self-conceit)." [Al-Bazzaar and Al-Bayhaqi - Al-Albaani classified it as Hasan (good)]

The Prophet also said, "If you were not committing sins, I would fear for you what is more serious than that: pride, pride." [Al-Bayhaqi - Al-Albaani classified it as Hasan]

Moreover, the Prophet said, "While a man was walking, clad in a two-piece garment and proud of himself with his hair well-groomed, Allah suddenly made him sink into the earth, and he will go on sinking into it till the Day of Resurrection." [Al-Bukhaari and Muslim]

Ibn Mas‘ood, may Allah be pleased with him, said, "There are two reasons for destruction: despair (of the Mercy of Allah) and pride."

Looking down upon people with contempt is considered forbidden arrogance, and every conduct that includes rejecting the truth or despising people is considered dispraised arrogance.

For more benefit, please refer to fatwas 173229 and 19854.

Allah knows best.