The Urdu and Persian word for animal sacrifice used is Qurbani (Sacrificial slaughter), it is derived from the Arabic word Qurban. Lexically, it means an act performed to seek Allah’s (Glory be to Him) pleasure. Originally, the word Qurban included all acts of charity because the purpose of charity is nothing but to seek Allah’s (Glory be to Him) pleasure. But, in precise religious terminology, the word was later confined to the sacrifice of an animal slaughtered for the sake of Allah (Glory be to Him).
Udhiyah refers to the animals of An’am class (camel, cow, goat or sheep) that is sacrificed as an act of worship to Allaah, during the period from after the Eid prayer on the Day of Nahr (Eid al-Adha) until the last day of Tashriq (the 13th day of Dhul-Hijjah), with the intention of offering sacrifice. Hady refers to the sacrifice offered by Hajj pilgrims in Makkah while Udhiyah refers to the sacrifice offered by non-pilgrims. 
The sacrifice of an animal has always been treated as a recognized form of worship in all religious orders originating from a divine book. Even in pagan societies, the sacrifice of an animal is recognized as a form of worship, but it is done in the name of some idols and not in the name of Allah (Glory be to Him), a practice totally rejected by Islam.
In the Shariah (Islamic Law) of our beloved Prophet Muhammad (May Allah honor Him and grant Him peace), the sacrifice of an animal has been recognized as a form of worship only during four days of Zulhijjah, namely, the 10th, 11th, 12th and 13th of the month, the day of al-Adha and the days of Tashreeq which are the three days after the day of al-Adha. ‘Ali ibn Abi Taalib (R) said: “The days of sacrifice are the Day of Sacrifice (yawm al-nahr) and the three days after it.” It is forbidden to fast on these days. The hadeeth was classed as saheeh by al-Albaani in al-Silsilah al-Saheehah, 2476 And ‘Aa’isha and Ibn ‘Umar (may Allah be pleased with them) said: “There is no concession allowing anyone to fast on the days of al-Tashreeq except the one who does not have an animal to sacrifice.” Sahih Al Bukhari 1884(NE) 
This is to commemorate the unparalleled sacrifice offered by the Prophet Ibrahim (Peace be upon Him), when he, in pursuance to a command of Allah (Glory be to Him) conveyed to him in a dream, prepared himself to slaughter his beloved son Ismail (Peace be upon Him) and actually did so but, Allah (Glory be to Him), after testing his submission, sent down a sheep and saved his son from the logical fate of slaughter. It is from that time onwards that the sacrifice of an animal became an obligatory duty to be performed by every well to do Muslim.
Qurbani is a demonstration of total submission to Allah (Glory be to Him) and a proof of complete obedience to Allah’s will or command. When a Muslim offers a Qurbani, this is exactly what he intends to prove. Thus, the Qurbani offered by a Muslim signifies that he is a slave of Allah (Glory be to Him) at his best and that he would not hesitate even for a moment, once he receives an absolute command from his Creator, to surrender before it, to obey it willingly, even if it be at the price of his life and possessions. When a true and perfect Muslim receives a command from Allah (Glory be to Him), he does not make his obedience dependent upon the command's reasonability' as perceived through his limited understanding. He knows that Allah (Glory be to Him) is All-knowing, All-Wise and that his own reason cannot encompass the knowledge and wisdom underlying the divine command. He, therefore, submits to the divine command, even if he cannot grasp the reason or wisdom behind it. 
Allah (Glory be to Him) said in many places that everything that a human does should be for the sake of Allah (Glory be to Him), even if you are sacrificing than that should be for the sake of Allah (Glory be to Him).Allah (Glory be to Him)said, "Say, 'My prayer, my sacrifice, indeed my whole life and death are to Allah, Lord of the universe, to whom there is no associate.'" Qur'an.Surah Anam 6:163-164 and "Address therefore your prayers to your Lord, and slaughter (in His name)" Qur'an.Surah Kauthar 108:2