Short answer
Allah sent prophets to every nation, and only He knows their exact number. The Quran names 25 of them, from Adam to Muhammad ﷺ. A well-known report gives 124,000 prophets, over 300 of them messengers, but Hanafi creed books advise not to fix a number, since the report is not strong enough for creed.
Belief in all the prophets in general is required, and in each prophet named in the Quran specifically. Following al-Nasafi's creed, the safer course is not to limit them to a number, so that no prophet is excluded and no non-prophet is included.
The four schools agree on this.
Explanation
The Quran says there was no nation without a warner (35:24), and that Allah told the Prophet ﷺ the stories of some messengers and not others (40:78, 4:164). So the prophets were many, sent to peoples all over the earth, and most of their names are not known to us.
The 25 named in the Quran are: Adam, Idris, Nuh, Hud, Salih, Ibrahim, Lut, Isma'il, Ishaq, Ya'qub, Yusuf, Ayyub, Shu'ayb, Musa, Harun, Dhul-Kifl, Dawud, Sulayman, Ilyas, al-Yasa', Yunus, Zakariyya, Yahya, Isa and Muhammad, peace be upon them all. Some scholars discuss whether Dhul-Kifl was a prophet or a righteous man; most count him as a prophet.
The figure of 124,000 prophets comes from reports of Abu Dharr and Abu Umama in the Musnad of Imam Ahmad and other books, which give the messengers as 313 or 315. Hadith scholars have discussed the strength of its chains. That is why al-Nasafi wrote in his creed that the better course is not to restrict them to a number.
A nabi (prophet) receives revelation from Allah. A rasul (messenger), in the common definition, is a prophet sent with a new law or book to a people. Every messenger is a prophet, but not every prophet is a messenger. Earlier prophets were sent to their own people, while the Prophet Muhammad ﷺ was sent to all of humanity, and he is the last of the prophets (33:40). Muslims make no distinction between them in belief (2:285). Read their stories on Stories of the Prophets.
The evidence
Quran, Surah Ghafir 40:78
وَلَقَدْ أَرْسَلْنَا رُسُلًۭا مِّن قَبْلِكَ مِنْهُم مَّن قَصَصْنَا عَلَيْكَ وَمِنْهُم مَّن لَّمْ نَقْصُصْ عَلَيْكَ ۗ وَمَا كَانَ لِرَسُولٍ أَن يَأْتِىَ بِـَٔايَةٍ إِلَّا بِإِذْنِ ٱللَّهِ ۚ فَإِذَا جَآءَ أَمْرُ ٱللَّهِ قُضِىَ بِٱلْحَقِّ وَخَسِرَ هُنَالِكَ ٱلْمُبْطِلُونَ ﴿٧٨﴾
And We have already sent messengers before you. Among them are those [whose stories] We have related to you, and among them are those [whose stories] We have not related to you. And it was not for any messenger to bring a sign [or verse] except by permission of Allah. So when the command of Allah comes, it will be concluded in truth, and the falsifiers will thereupon lose [all].
Not every prophet is named in the Quran.
Quran, Surah Faatir 35:24
إِنَّآ أَرْسَلْنَٰكَ بِٱلْحَقِّ بَشِيرًۭا وَنَذِيرًۭا ۚ وَإِن مِّنْ أُمَّةٍ إِلَّا خَلَا فِيهَا نَذِيرٌۭ ﴿٢٤﴾
Indeed, We have sent you with the truth as a bringer of good tidings and a warner. And there was no nation but that there had passed within it a warner.
Every nation received a warner.
Quran, Surah Al-Ahzaab 33:40
مَّا كَانَ مُحَمَّدٌ أَبَآ أَحَدٍۢ مِّن رِّجَالِكُمْ وَلَٰكِن رَّسُولَ ٱللَّهِ وَخَاتَمَ ٱلنَّبِيِّۦنَ ۗ وَكَانَ ٱللَّهُ بِكُلِّ شَىْءٍ عَلِيمًۭا ﴿٤٠﴾
Muhammad is not the father of [any] one of your men, but [he is] the Messenger of Allah and last of the prophets. And ever is Allah, of all things, Knowing.
Muhammad is the last prophet.
Sahih al-Bukhari 335
Sahih... وَكَانَ النَّبِيُّ يُبْعَثُ إِلَى قَوْمِهِ خَاصَّةً وَبُعِثْتُ إِلَى النَّاسِ عَامَّةً ...
"Every prophet was sent to his own people only, and I have been sent to all people."
Shows earlier prophets were sent to their own nations, and the Prophet to everyone.
Where the ruling is recorded
- al-Aqa'id al-Nasafiyya with Sharh al-Taftazani, Section on the prophets. States that a number is reported in some hadith, but it is better not to restrict them to a number.
- al-Fiqh al-Akbar (Imam Abu Hanifa), Section on the prophets. Affirms belief in all the prophets and their protection from sin.
This answer is pending review by the scholars on our team. 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.