Short answer
Bin (or ibn) means "son of" and bint means "daughter of". Muhammad bin Abdullah is Muhammad, son of Abdullah. Using it is a tradition, not an obligation, but Islam does require that a person is never linked to anyone other than their real father.
Calling a child by the name of their true father is wajib, and falsely claiming another father is haram. The form of the name, whether bin, a family surname or a Western last name, is a matter of custom and is permissible.
The four schools agree on this.
Explanation
In Arabic, a traditional name is a chain. Ibn means son, and it is usually written bin when it falls between two names, as in Umar bin al-Khattab. Bint means daughter, as in Fatimah bint Muhammad. Other parts of a classical name are the kunyah, Abu (father of) or Umm (mother of), as in Abu Bakr or Umm Salamah, and the nisbah, which links a person to a place or tribe, such as al-Bukhari, from Bukhara.
Why it matters in fiqh: the four schools agree that lineage is fixed by Allah and cannot be changed. Allah says to call children by the names of their fathers. The Prophet ﷺ warned that whoever knowingly claims someone other than his father has committed a grave sin. This is why Islam keeps adoption and step-parenting from changing a child's name; see can an adopted child take your last name.
Do you have to use bin? No. A family surname, as is normal in the US, the UK and South Asia, is permissible. Many Muslims in the Gulf use bin or Al (family of) in official names, while others drop it. A married woman traditionally keeps her father's name, since marriage does not change lineage; on taking a husband's surname, see changing your last name.
For names and their meanings, see the Islamic names list.
The evidence
Quran, Surah Al-Ahzaab 33:5
ٱدْعُوهُمْ لِءَابَآئِهِمْ هُوَ أَقْسَطُ عِندَ ٱللَّهِ ۚ فَإِن لَّمْ تَعْلَمُوٓا۟ ءَابَآءَهُمْ فَإِخْوَٰنُكُمْ فِى ٱلدِّينِ وَمَوَٰلِيكُمْ ۚ وَلَيْسَ عَلَيْكُمْ جُنَاحٌۭ فِيمَآ أَخْطَأْتُم بِهِۦ وَلَٰكِن مَّا تَعَمَّدَتْ قُلُوبُكُمْ ۚ وَكَانَ ٱللَّهُ غَفُورًۭا رَّحِيمًا ﴿٥﴾
Call them by [the names of] their fathers; it is more just in the sight of Allah. But if you do not know their fathers - then they are [still] your brothers in religion and those entrusted to you. And there is no blame upon you for that in which you have erred but [only for] what your hearts intended. And ever is Allah Forgiving and Merciful.
Commands that every person be named after their true father.
Sahih al-Bukhari 3508
Sahih... ليس من رجل ادعى لغير أبيه وهو يعلمه إلا كفر ...
"There is no man who knowingly claims someone other than his father except that he has committed an act of disbelief."
Shows how serious it is to falsify lineage.
Where the ruling is recorded
- al-Fatawa al-Hindiyya, Kitab al-Karahiyya, chapter on names and kunyas. Hanafi rulings on names and kunyas.
- Radd al-Muhtar (Ibn Abidin), Kitab al-Hazr wa al-Ibahah, discussion of names. On naming and the kunyah.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.