Although these three things are used to identify a person, they achieve this differently:

Name

The word that is used to identify you from birth and does not usually contain the word Ab or Umm or any form of praise or insult, like ʿAbd al-Raḥmān, Ḥamzah, Yaḥyá, ʿĀʾishah, Laylá.

Laqab (Nickname)

Any word that usually succeeds your name in Arabic and contains some sort of praise or insult, like tall in the laqab tall Uways [i.e. Uways al-Ṭawīl], and blind in the laqab Blind Muḥammad or Muḥammad the Blind [i.e. Muḥammad al-Aʿmash]. Utilising the first type of laqab is allowed; however, utilising an insulting laqab is generally prohibited. Allāh said:

“وَلَا تَنَابَزُوا بِالْأَلْقَابِ”
“Do not address each other utilising offensive nicknames.”
[al-Ḥujarāt, 49:11]

If the user intends to insult the person, then utilising an offensive laqab is prohibited; however, if the user intends to simply identify which person he is speaking about without insulting him—like Blind Muḥammad or Muḥammad the Blind [i.e. Muḥammad al-Aʿmash] if both individuals know multiple people named Muḥammad—then this is allowed, as the intent here is not to insult or ridicule the person.

Therefore, there are three types of nicknames:

1. The type that is used to praise, exalt and honour someone, like King Salmān [i.e. Al-Malik Salmān]. Utilising this type of laqab is undoubtedly allowed.
2. The type that can be construed to be insulting and degrading but is used to identify someone, like Crippled Khālid or Khālid the Cripple [i.e. Khālid al-Aʿraj]. Utilising this type of laqab is also allowed.
3. The insulting and humiliating type that is used to insult, humiliate and degrade someone, like “ʿAlī the Donkey” [i.e. ʿAlī al-Ḥimār]. Utilising this type of laqab is unquestionably prohibited.

Al-Fayrūzābādī (816 AH)—a renowned lexicographer—said:

There are three types of nicknames:

1. The type that is used to praise someone
2. The type that is used to identify someone
3. The type that is used to insult someone
   (Baṣāʾir Dhwī al-Tamyīz, vol. 4, pg. 438)

Kunyah

Any name that starts with Ab or Umm [i.e father or mother], like Abū Hurayrah, Abū Ḥakīm [i.e. Ḥakīm’s father], Abū Khadījah [i.e. Khadījah’s father], Umm Ṣāliḥ [i.e. Ṣāliḥ’s mother], Umm Ḥafṣah [i.e. Ḥafṣah’s mother], and whenever the Arabs wanted to exalt and honour someone, they would address them by their kunyah rather then addressing them by their name or their nickname. This is why some scholars are of the opinion that a kāfir should not be given a kunyah. It is important to note that a kunyah can sometimes be used as a laqab and vice-versa, like Abū al-Jūd and Abū Lahab.

Therefore, a person can be identified by their given name, or by a defining characteristic, or by their children in the case of parents.