Qurban - Difference Between Qurban, Aqīqah & Hadyu

Qurban, Aqīqah and Hadyu are terms that entail sacrificial rituals. They can be distinguished in terms of aḥkām (rulings) and definitions. 

Qurban:

Qurban is the sacrificial ritual on the *specific days of Zulhijjah (from the 10th to 13th)*. It is *Sunnah Muakkadah* (highly recommended) 

Aqiqah:

Aqīqah is the sacrifice made *to celebrate the birth of a child*. It is not specific to the mentioned days of Zulhijjah as per the act of Qurban. In fact, it can be *performed in any other days of the year*. The *best day to perform ‘Aqīqah, however, is on the seventh day after birth*. The Prophet s.a.w. said:  

حَدَّثَنَا عَلِيُّ بْنُ حُجْرٍ، أَخْبَرَنَا عَلِيُّ بْنُ مُسْهِرٍ، عَنْ إِسْمَاعِيلَ بْنِ مُسْلِمٍ، عَنِ الْحَسَنِ، عَنْ سَمُرَةَ، قَالَ قَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم ‏ "‏ الْغُلاَمُ مُرْتَهَنٌ بِعَقِيقَتِهِ يُذْبَحُ عَنْهُ يَوْمَ السَّابِعِ وَيُسَمَّى وَيُحْلَقُ رَأْسُهُ ‏"‏ ‏.‏

Narrated Samurah:

That the Messenger of Allah (ﷺ) said: "The boy is mortgaged by his *'Aqiqah; slaughtering should be done for him on the seventh day, he should be given a name, and his head should be shaved.*"

Jami` at-Tirmidhi 1522

Grade: Sahih (Darussalam)

https://sunnah.com/tirmidhi:1522

Other references:

https://sunnah.com/ibnmajah:3165 (Grade Hasan)

https://sunnah.com/nasai:4220 (Grade Hasan) 

But if it is not possible for parents to do it on the seventh day, it can still be done in any other time *before the child reaches the age of puberty*. It is *sunnah to sacrifice 2 goats for a Baby Boy and 1 goat for a Baby Girl.* 

Hadyu:

Hadyu is the one *performed on Yawm An-Naḥr (literally, day of sacrifice i.e. Aidiladha)* by the *Pilgrims in particular*. Hadyu is generally *Wājib (obligatory) upon *Pilgrims who may have met certain conditions during Hajj or Umrah, or did not perform any of the Manasik (integral rituals of Hajj)* such as not making the intention at Mīqāt (a boundary where each pilgrim makes their intention to perform the Hajj or Umrah).