It means to proclaim oneself as the great and superior. In Urdu the word 'takabar' is used for arrogance. By studying Holy Quran, it is known that satan did arrogance first among creatures and said, "I am better than Adam. Therefore, I will not prostrate in front of him".
Allah said in response to him.
قَالَ فَاهْبِطُ مِنْهَا فَمَا يَكُونُ لَكَ اَنْ تَتَكَبَّرَ فِيهَا فَاخْرُجُ إِنَّكَ مِنَ الصغيرين (اعراف : 13)
Translation: (Allah) said, "Descend from paradise, for it is not for you to be arrogant therein. So get out, indeed, you are of the debased".
i. The First Arrogant:
The first one among creature, who did arrogance is Iblis (Satan). Allah s commanded him to prostrate and then he said with pride.
أَنَا خَيْرُ مِنْهُ خَلَقْتَنِي مِنْ نَارٍ وَخَلَقْهُ مِنْ طِينٍ . (الاعراف : 12)
Translation: (Satan) said, I am better than him. You created me from fire and created him from clay.
Satan went away from mercy of Allah due to his arrogance as it is said by Allah
ابى وَاسْتَكْبَرَ وَكَانَ مِنَ الكَفِرِينَ) (البقرة:34)
Translation: He refused and was arrogant and became of the disbelievers.
ii. Arrogance in light of Quran:
Arrogance is condemned in Holy Quran. It is said by Allah]
اليس في جَعَلَمَ مَثْوَى لِلْمُتَكَفِرِينَ (الزمر: 60)
Translation: Is there not in hell, a residence for the arrogant?
إن الله لا يُحِبُّ كُل مُختال فخورة (لقمان: 18)
Translation: Indeed, Allah does not like everyone self-deluded and boastful.
الله لا يُحِبُّ الْمُسْتَكْبِرِينَ (الحل:23)
Translation: Indeed, he does not like the arrogant.
iii. Arrogance in light of hadees:
لا يَدْخُلُ الْجَنَّةَ مَنْ كَانَ فِي قَلْبِهِ مِثْقَالُ الرَّةٍ مِنْ كَبُرٍ
Translation: The person who has arrogance (even) equal to a particle in his heart, he will not enter paradise.
The Holy Prophet said that in ancient times a person wearing dress came out while doing arrogance, Allah commanded earth to take him in its grasp, so the earth caught him. Then he would be thirst into earth till the day of judgement.
An arrogant person does cruelty while thinking others inferior and becomes bold on committing sins. He thinks who could punish him over his sins. Therefore, he is deprived of affection, brotherhood, sacrifice and many such welfare traits.