First, let us understand what Udhiyah stands for. The word Udhiyah means an animal of the An’aam class (camel, cow, sheep or goat) that is slaughtered during the days of Eid-ul-Adha as an act of worship, intending to draw closer to Allah thereby.
One who wants to slaughter the Udhiyah has to say Bismillaah wa Allaahu akbar, Allaahumma haaza minka wa laka, haaza ‘anni (or if it is being offered on behalf of someone else, haaza ‘an [fulaan]), Allaahumma taqabbal min [fulaan] wa aali [fulaan]. (In the name of Allah, Allah is most great. O Allah, this is from You and to You. This is on my behalf (or if it is being offered on behalf of someone else, this is on behalf of [So and so]). O Allah, accept (this sacrifice) from [So and so] and the family of [So and so] – here one should mention his name instead of [fulaan] or [So and so].
Allaahumma minka (O Allaah, from You) means this sacrifice is a gift and provision that has reached me from You. Laka (to You) means, it is sincerely for You alone.
However, what is obligatory here is to say Bismillaah; the rest is mustahabb.
Anas RA said: The Prophet PBUH sacrificed two horned rams that were white speckled with black. He slaughtered them with his own hand, said Bismillaah and Allaahu akbar, and put his foot on their necks (Bukhari and Muslim).
Aaishah RA narrets that Prophet PBUH ordered that a horned ram be brought to him so that he could sacrifice it. He said, “O ‘Aa’ishah, give me the knife.” Then he said, “Sharpen it on a stone.” So she did that, then he took it and took the ram, and he lay it down and prepared to slaughter it. He said, “In the name of Allah, O Allah accept (this sacrifice) from Muhammad and the family of Muhammad and the ummah of Muhammad,” then he sacrificed it, (Muslim).
Jabir bin Abdullah said: I was present with the Prophet PBUH on the day of Eid-ul-Adha at the prayer place. When he had finished his Khutbah he came down from his minbar (pulpit) and a ram was brought which the Messenger of Allaah PBUH slaughtered with his own hand. He said, “In the name of Allah, Allah is Most Great. This is on behalf of myself and on behalf of those of my ummah who have not offered a sacrifice.” Classed as saheeh by al-Albaani in Saheeh al-Tirmidhi.