Allah (God) had sent many Rusul (Messengers) to this earth for the guidance of mankind. Names of only twenty-five Rusul (Messengers) are mentioned in the Holy Qur-an.
Nuh عليه السلام (Noah) had a especial place among the Rusul (Messengers) of Allah (God).

إِنَّ اللَّهَ اصْطَفَىٰ آدَمَ وَنُوحًا وَآلَ إِبْرَاهِيمَ وَآلَ عِمْرَانَ عَلَى الْعَالَمِينَ
Allah Chose Adam, Nuh (Noah), the family of Ibrahim (Abraham) and the family of ‘Imran above the ‘Alamin (mankind and jinn) (of their times).
Surah (Chapter) Al-’Imran (The Family of ‘Imran) 3:33

Allah (God) had sent Nuh عليه السلام (Noah) with His Guidance to mankind before Ibrahim عليه السلام (Abraham).

إِنَّا أَوْحَيْنَا إِلَيْكَ كَمَا أَوْحَيْنَا إِلَىٰ نُوحٍ وَالنَّبِيِّينَ مِن بَعْدِهِ ۚ وَأَوْحَيْنَا إِلَىٰ إِبْرَاهِيمَ وَإِسْمَاعِيلَ وَإِسْحَاقَ وَيَعْقُوبَ وَالْأَسْبَاطِ وَعِيسَىٰ وَأَيُّوبَ وَيُونُسَ وَهَارُونَ وَسُلَيْمَانَ ۚ وَآتَيْنَا دَاوُودَ زَبُورًا
Verily, We have sent the revelation to you (O Muhammad S) as We sent the revelation to Nuh (Noah) and the Prophets after him; We also sent the revelation to Ibrahim (Abraham), Isma’il (Ishmael), Ishaq (Isaac), Ya’qub (Jacob), and Al-Asbat [the offspring of the twelve sons of Ya’qub (Jacob)], ‘Isa (Jesus), Ayyub (Job), Yunus (Jonah), Harun (Aaron), and Sulayman (Solomon); and to Dawud (David) We gave the Zabur (Psalms).
Surah (Chapter) An- Nisa” (The Women) 4:163

An entire Surah (Chapter) of the Holy Qur-an, its 71st chapter, has been named after him. Besides Surah (Chapter) Nuh, portions of his story are mentioned in many other places in the Holy
Qur-an. Nuh عليه السلام (Noah) is mentioned by name in the following Ayah (Verses) of the Holy Qur-an:

3:33, 4:163, 6:84, 7:59-69, 9:70, 10:71-73, 11:25,32,36,42,45,46,48,89, 17:3, 21:76, 23:23, 25:37, 26:105, 28:14, 37:75, 51:46, 54:9, 69:11, 71:1-28

People of Nuh عليه السلام (Noah) were mostly rich who were used to the luxuries of life. Their riches had made them proud and arrogant, the two qualities detested by Allah (God). They also indulged in idol worship, the worst form of Shirk (Taking partners with Allah) that Allah (God) would never forgive.

People of Nuh عليه السلام (Noah) had assigned many more gods for themselves besides worshipping the idols of Wadd, Suwa”, Yaghuth, Ya’uq and Nasr.

َقَالُوا لَا تَذَرُنَّ آلِهَتَكُمْ وَلَا تَذَرُنَّ وَدًّا وَلَا سُوَاعًا وَلَا يَغُوثَ وَيَعُوقَ وَنَسْرًا
“And they have said: ‘You shall not leave your gods: nor shall You leave Wadd, nor Suwa”, nor Yaghuth, nor Ya’uq nor Nasr. (these are the names of their idols).
Surah (Chapter) Nuh (Noah) 71:23

Allah (God) decided to make Nuh عليه السلام (Noah), a poor man, as His Rasul (Messenger) to warn those arrogant idol worshippers.

لَقَدْ أَرْسَلْنَا نُوحًا إِلَىٰ قَوْمِهِ فَقَالَ يَا قَوْمِ اعْبُدُوا اللَّهَ مَا لَكُم مِّنْ إِلَـٰهٍ غَيْرُهُ إِنِّي أَخَافُ عَلَيْكُمْ عَذَابَ يَوْمٍ عَظِيمٍ
Indeed, We sent Nuh (Noah) to his people and he said: “O my people! Worship Allah! You have no other ilah (god) but Him. (La ilaha illallah: none has the right to be worshipped but Allah). Certainly I fear for you the torment of a Great Day!”
Surah (Chapter) al-A’raf (The Heights) 7:59

Nuh عليه السلام (Noah) told his people that Allah (God) will forgive their previous sins if they do Tawbah (Repentance) sincerely from their sins and spend the rest of their lives according to the way that is liked by Him.

Like most Rusul (Messengers) of Allah (God), Nuh عليه السلام (Noah) also struggled and suffered throughout his life for taking the Message of Allah (God) to his people. They did not listen to him and ridiculed and tortured him for preaching the Truth.

People of Nuh عليه السلام (Noah) could not comprehend his special stature and they dismissed his invitations to the Path of Allah (God) for the simple reason that he was also a human being who had spent all his life with them.
His preaching attracted mostly the poor and the downtrodden to the Message of Allah (God). This fact also became a tool for the people of Nuh عليه السلام (Noah) to make fun of him and ridicule him further.

فَقَالَ الْمَلَأُ الَّذِينَ كَفَرُوا مِن قَوْمِهِ مَا نَرَاكَ إِلَّا بَشَرًا مِّثْلَنَا وَمَا نَرَاكَ اتَّبَعَكَ إِلَّا الَّذِينَ هُمْ أَرَاذِلُنَا بَادِيَ الرَّأْيِ وَمَا نَرَىٰ لَكُمْ عَلَيْنَا مِن فَضْلٍ بَلْ نَظُنُّكُمْ كَاذِبِينَ
The chiefs who disbelieved among his people said: “We see you but a man like ourselves, nor do we see any follow you but the meanest among us and they (too) followed without thinking. And we do not see in you any merit above us, infact we think you are liars.”
Surah (Chapter) Hud 11:27

People of Nuh عليه السلام (Noah) were skeptical of his claims of Messengerhood, because they did not see any things special in him. They were convinced that either he is doing it for selfish purposes to propel himself up among them or he had gone crazy.

فَقَالَ الْمَلَأُ الَّذِينَ كَفَرُوا مِن قَوْمِهِ مَا هَـٰذَا إِلَّا بَشَرٌ مِّثْلُكُمْ يُرِيدُ أَن يَتَفَضَّلَ عَلَيْكُمْ وَلَوْ شَاءَ اللَّهُ لَأَنزَلَ مَلَائِكَةً مَّا سَمِعْنَا بِهَـٰذَا فِي آبَائِنَا الْأَوَّلِينَ
But the chiefs of his people who disbelieved said: “He is no more than a human being like you, he seeks to make himself superior to you. Had Allah willed, He surely could have sent down angels. Never did we hear such a thing among our fathers of old.
Surah (Chapter) al-Mu”minun (The Believers) 23:24

إِنْ هُوَ إِلَّا رَجُلٌ بِهِ جِنَّةٌ فَتَرَبَّصُوا بِهِ حَتَّىٰ حِينٍ
“He is only a man in whom is madness, so wait for him a while.”
Surah (Chapter) al-Mu”minun (The Believers) 23:25

Nuh عليه السلام (Noah) did not mind those baseless accusations and insult and continued begging his people to listen to the Message of Allah (God) and mend their ways. He told them that he is inviting them to the path of Allah (God) for their own good and had no selfish motive behind it.

يَا قَوْمِ لَا أَسْأَلُكُمْ عَلَيْهِ أَجْرًا ۖ إِنْ أَجْرِيَ إِلَّا عَلَى الَّذِي فَطَرَنِي ۚ أَفَلَا تَعْقِلُونَ
“O my people I ask of you no reward for it (the Message). My reward is only from Him Who created me. Will you not then understand?
Surah (Chapter) Hud 11:51

Nuh عليه السلام (Noah) persisted for 950 years in inviting his people to the path of Allah (God).

وَلَقَدْ أَرْسَلْنَا نُوحًا إِلَىٰ قَوْمِهِ فَلَبِثَ فِيهِمْ أَلْفَ سَنَةٍ إِلَّا خَمْسِينَ عَامًا فَأَخَذَهُمُ الطُّوفَانُ وَهُمْ ظَالِمُونَ
And indeed We sent Nuh (Noah) to his people, and he stayed among them a thousand years less fifty years [inviting them to believe I the Oneness of Allah (Monotheism), and discard the false gods and other desires]; and the deluge overtook them while they were Zalimun (wrong-doers, polytheists, disbelievers).
Surah (Chapter) al-‘Ankabut (The Spider) 29:14

Nuh عليه السلام (Noah) told Allah (God) that he had sincerely done his job but to no avail.

قَالَ رَبِّ إِنِّي دَعَوْتُ قَوْمِي لَيْلًا وَنَهَارًا
He said: “O my Lord! Verily, I have called to my people night and day (i.e. secretly and openly to accept the doctrine of Monotheism),
فَلَمْ يَزِدْهُمْ دُعَائِي إِلَّا فِرَارًا
“But all my calling added nothing but to (their) flight (from the truth).
وَإِنِّي كُلَّمَا دَعَوْتُهُمْ لِتَغْفِرَ لَهُمْ جَعَلُوا أَصَابِعَهُمْ فِي آذَانِهِمْ وَاسْتَغْشَوْا ثِيَابَهُمْ وَأَصَرُّوا وَاسْتَكْبَرُوا اسْتِكْبَارًا
And verily, every time I called unto them that you might forgive them, they thrust their fingers into their ears, covered themselves up with their garments, and persisted (in their refusal), and magnified themselves in pride.
Surah (Chapter) Nuh (Noah) 71:5-7

People of Nuh عليه السلام (Noah) never took him seriously and dismissed his preaching and warnings. They even challenged him to get that torment for them that he used to mention.

قَالُوا يَا نُوحُ قَدْ جَادَلْتَنَا فَأَكْثَرْتَ جِدَالَنَا فَأْتِنَا بِمَا تَعِدُنَا إِن كُنتَ مِنَ الصَّادِقِينَ
They said: “O Nuh (Noah)! You have disputed with us and much have you prolonged this dispute with us, now bring upon us that you threaten us with, if you are of the truthful.”
Surah (Chapter) Hud 11:32

Nuh عليه السلام (Noah) told them that he does not know about future, and only Allah (God) knows about the time and type of punishment that He will bring upon them:

قَالَ إِنَّمَا يَأْتِيكُم بِهِ اللَّهُ إِن شَاءَ وَمَا أَنتُم بِمُعْجِزِينَ
He said: “Only Allah will bring it (the punishment) on you, if He wills, and then you will escape not.
Surah (Chapter) Hud 11:33

Despite of telling them about the great punishment Allah (God) will bring upon his people, Nuh عليه السلام (Noah) was concerned about them. He finally told his people that he would not be able to help them anymore, because Allah (God) had revealed to him that none of his disbelieving people would now listen to him.  

وَأُوحِيَ إِلَىٰ نُوحٍ أَنَّهُ لَن يُؤْمِنَ مِن قَوْمِكَ إِلَّا مَن قَدْ آمَنَ فَلَا تَبْتَئِسْ بِمَا كَانُوا يَفْعَلُونَ
And it was revealed to Nuh (Noah): “None of your people will believe except those who have believed already. So be not sad because of what they used to do.
Surah (Chapter) Hud 11:36

Allah (God) commanded Nuh عليه السلام (Noah) to construct a ship.

وَاصْنَعِ الْفُلْكَ بِأَعْيُنِنَا وَوَحْيِنَا وَلَا تُخَاطِبْنِي فِي الَّذِينَ ظَلَمُوا ۚ إِنَّهُم مُّغْرَقُونَ
“And construct the ship under Our Eyes and with Our Revelation, and call not upon Me on behalf of those who did wrong; they are surely to be drowned.”
Surah (Chapter) Hud 11:37

Being an obedient servant of Allah (God), Nuh عليه السلام (Noah) listened to His command and immediately began constructing his ark, the greatest ship ever built. His people poked fun at him seeing him toil for building that ship on dry land far away from any water body. They asked him that how would he take this huge ship to water.

Nuh A (Noah) patiently listened to their ridicules and continued building his ark. He told his people that they are the ones who will be finally disgraced.

And as he was constructing the ship, whenever the chiefs of his people passed by him, they mocked at him. He said: “If you mock at us, so do we mock at you likewise for your mocking.

وَيَصْنَعُ الْفُلْكَ وَكُلَّمَا مَرَّ عَلَيْهِ مَلَأٌ مِّن قَوْمِهِ سَخِرُوا مِنْهُ ۚ قَالَ إِن تَسْخَرُوا مِنَّا فَإِنَّا نَسْخَرُ مِنكُمْ كَمَا تَسْخَرُونَ
“And as he was constructing the ship, whenever the chiefs of his people passed by him, they mocked at him. He said: "If you mock at us, so do we mock at you likewise for your mocking.
فَسَوْفَ تَعْلَمُونَ مَن يَأْتِيهِ عَذَابٌ يُخْزِيهِ وَيَحِلُّ عَلَيْهِ عَذَابٌ مُّقِيمٌ
“And you will know who it is on whom will come a torment that will cover him with disgrace and on whom will fall a lasting torment.
Surah (Chapter) Hud 11:38-39

When Nuh عليه السلام (Noah) completed his ark, Allah (God) commanded him to take a male and a female from all species of animals, birds and insects; besides his believing followers.

حَتَّىٰ إِذَا جَاءَ أَمْرُنَا وَفَارَ التَّنُّورُ قُلْنَا احْمِلْ فِيهَا مِن كُلٍّ زَوْجَيْنِ اثْنَيْنِ وَأَهْلَكَ إِلَّا مَن سَبَقَ عَلَيْهِ الْقَوْلُ وَمَنْ آمَنَ ۚ وَمَا آمَنَ مَعَهُ إِلَّا قَلِيلٌ
(So it was) till when Our Command came and the oven gushed forth (water like fountains from the earth). We said: “Embark therein, of each kind two (male and female), and your family-except him against whom the Word has already gone forth- and those who believe. And none believed with him except a few.”
Surah (Chapter) Hud 11:40

Son of Nuh عليه السلام (Noah) was also among the unbelievers. Nuh عليه السلام (Noah) knew that his son will also be punished with the unbelievers; still he pleaded with Allah (God) to save his son citing that he also belongs to his family.

وَنَادَىٰ نُوحٌ رَّبَّهُ فَقَالَ رَبِّ إِنَّ ابْنِي مِنْ أَهْلِي وَإِنَّ وَعْدَكَ الْحَقُّ وَأَنتَ أَحْكَمُ الْحَاكِمِينَ
And Nuh (Noah) called upon his Lord and said, “O my Lord! Verily, my son is of my family! And certainly, your promise is true, and you are the Most Just of the judges.”
Surah (Chapter) Hud 11:45

Allah (God) corrected Nuh عليه السلام (Noah) with a stern warning.

قَالَ يَا نُوحُ إِنَّهُ لَيْسَ مِنْ أَهْلِكَ ۖ إِنَّهُ عَمَلٌ غَيْرُ صَالِحٍ ۖ فَلَا تَسْأَلْنِ مَا لَيْسَ لَكَ بِهِ عِلْمٌ ۖ إِنِّي أَعِظُكَ أَن تَكُونَ مِنَ الْجَاهِلِينَ
He said: “O Nuh (Noah)! Surely, he is not of your family; verily his work is unrighteous, so ask not of Me that of which you have no knowledge! I admonish you lest you should be one of the ignorant.”
Surah (Chapter) Hud 11:46

Nuh عليه السلام (Noah) apologized to Allah (God) for his mistake of asking for Him for His forgiveness of his son, a disbeliever.

قَالَ رَبِّ إِنِّي أَعُوذُ بِكَ أَنْ أَسْأَلَكَ مَا لَيْسَ لِي بِهِ عِلْمٌ ۖ وَإِلَّا تَغْفِرْ لِي وَتَرْحَمْنِي أَكُن مِّنَ الْخَاسِرِينَ
Nuh (Noah) said: “O my Lord! I seek refuge with You from asking You that of which I have no knowledge. And unless You forgive me and have Mercy on me, I will indeed be one of the losers.”
Surah (Chapter) Hud 11:47

Allah (God) brought upon the storm that He had predicted to Nuh عليه السلام (Noah). Soon water reached to such level that the ark Nuh عليه السلام (Noah) had built began to sail. He knew that Allah (God) is soon going to unleash his torment fully, upon his unbelieving people. He called his son, one of the unbelievers, to believe in Allah (God) and board the ark.

وَهِيَ تَجْرِي بِهِمْ فِي مَوْجٍ كَالْجِبَالِ وَنَادَىٰ نُوحٌ ابْنَهُ وَكَانَ فِي مَعْزِلٍ يَا بُنَيَّ ارْكَب مَّعَنَا وَلَا تَكُن مَّعَ الْكَافِرِينَ
So it (the ship) sailed with them amidst waves like mountains, and Nuh (Noah) called out to his son, who had separated himself (apart): “O my son! Embark with us and be not with the disbelievers.”
Surah (Chapter) Hud 11:42

The son of  Nuh عليه السلام (Noah) and all other disbelievers thought that they would climb up on some mountain to save themselves from water, without realizing the magnitude of the torment that was about to befall upon them as a punishment for their misdeeds. Little did he know that no one can hide from Allah (God).

قَالَ سَآوِي إِلَىٰ جَبَلٍ يَعْصِمُنِي مِنَ الْمَاءِ ۚ قَالَ لَا عَاصِمَ الْيَوْمَ مِنْ أَمْرِ اللَّهِ إِلَّا مَن رَّحِمَ ۚ وَحَالَ بَيْنَهُمَا الْمَوْجُ فَكَانَ مِنَ الْمُغْرَقِينَ
(The son) replied: “I will betake myself to some mountain; it will save me from the water.” Nuh (Noah) said: “This day there is no savior from the decree of Allah except him on whom He has mercy.” And waves came between them, so he (the son) was among the drowned.
Surah (Chapter) Hud 11:43

The storm raged with unprecedented fury and soon drowned everything. Only the ark that Nuh عليه السلام (Noah) had constructed was left floating with everyone on board. Then Allah (God) stopped the storm.

وَقِيلَ يَا أَرْضُ ابْلَعِي مَاءَكِ وَيَا سَمَاءُ أَقْلِعِي وَغِيضَ الْمَاءُ وَقُضِيَ الْأَمْرُ وَاسْتَوَتْ عَلَى الْجُودِيِّ ۖ وَقِيلَ بُعْدًا لِّلْقَوْمِ الظَّالِمِينَ
And it was said: “O earth! Swallow up your water, and O sky! Withhold (your rain).” And the water was made to subside and the Decree (of Allah) was fulfilled (i.e. the destruction of the people of Nuh (Noah). And it (the ship) rested on (Mount) Judi, and it was said: “Away with the people who are Zalimun (polytheists and wrong-doers)!”
Surah (Chapter) Hud 11:44

عليه السلام - 'Alayhis Salam - A Du'a meaning 'Peace be upon him' is to be recited when the name of any Messenger of Allah (other than Muhammad صلى الله عليه وسلم) is mentioned.