What constitutes the neighbour in Islam? Are they only those who live closest to you, or does it cover everybody in your whole neighbourhood? What are the rights of the neighbours? Please clarify.

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.

One's neighbors include all the houses close by, within a perimeter of forty houses. Allah, The Exalted, says (what means): {Worship Allah and associate nothing with Him, and to parents do good, and to relatives, orphans, the needy, the near neighbor, the neighbor farther away, the companion at your side...} [Quran 4:36]

ʻAa'ishah said to the Prophet, sallallaahu ʻalayhi wa sallam, “O Messenger of Allah! I have two neighbors; which of them should I give a gift to?” The Prophet, sallallaahu ʻalayhi wa sallam, said, “(Give) to the one whose door is nearer to you.” [Al-Bukhari]

Ibn Hajar wrote:

“The scholars held different views regarding the perimeter of one’s neighborhood. It has been narrated on the authority of ʻAli  that all those who can hear the athaan of the mosque are neighbors. It has also been said that all those who pray the Fajr (dawn) prayer with you in a mosque are your neighbors. Also, ʻAa'ishah narrated that the perimeter of one’s neighborhood is forty houses on each side; the same view was adopted by Al-Awzaʻi. Al-Bukhari cited a marfooʻ (attributed to the Prophet, sallallaahu ‘alayhi wa sallam) hadeeth narrated by Kaʻb ibn Maalik with a weak chain of narration in his book Al-Adab Al-Mufrad, and it was also cited by Al-Hasan and At-Tabaraani that the perimeter of one’s neighborhood is forty houses. Ibn Wahb narrated on the authority of Yoonus ibn Shihaab that it is forty houses from every side; to the right, to the left, to the back and to the front (in which case one’s neighbors will amount to 160 or so). This view is possibly correct, just like the first one; it may also mean that it is forty houses in total, that is, ten houses from each side...” [Fat-h Al-Baari: 10/447]

For further benefit, please refer to fatwa 291052.

As for the rights of the neighbor in Islam, please refer to fatwa 85376.

Allah knows best.