The eclipse prayer is two rak'ahs, each with one bowing and two prostrations, with the recitation not said aloud

Al-Aswad bin Qays narrates from Tha'labah bin Abbad al-Abdi, one of the people of Basrah, that one day he heard a sermon of Sayyiduna Samurah bin Jundub (may Allah be pleased with him). In his sermon Sayyiduna Samurah (may Allah be pleased with him) said: One day, in the time of the Messenger of Allah (peace and blessings be upon him), a boy from the Ansar and I were shooting arrows at a target of ours, until, when the sun was two or three spear-lengths above the horizon in the eye of the beholder, it turned black until it became like a tannumah (a black-coloured plant). One of us said to the other: "Let us go to the mosque, for by Allah, this matter of the sun will surely bring about some new event for the Messenger of Allah (peace and blessings be upon him) in his Ummah." So we hurried to the mosque, and he (peace and blessings be upon him) had (already) come out (of his chamber), and we joined the Messenger of Allah (peace and blessings be upon him) as he came out to the people. The narrator said: He (peace and blessings be upon him) went forward and led us in a prayer longer than any he had ever led us in before, and we could not hear his (peace and blessings be upon him) voice. Then he (peace and blessings be upon him) bowed with us for longer than he had ever bowed with us before, and we could not hear his (peace and blessings be upon him) voice. Then he (peace and blessings be upon him) prostrated with us for longer than he had ever prostrated with us before, and we could not hear his (peace and blessings be upon him) voice. The narrator said: Then he (peace and blessings be upon him) did the same in the second rak'ah. The narrator said: The clearing of the sun coincided with his (peace and blessings be upon him) sitting in the second rak'ah. The narrator said: Then he (peace and blessings be upon him) gave the salam, praised and extolled Allah, bore witness that there is no god but Allah, and bore witness that he (peace and blessings be upon him) is His servant and Messenger. Then he said: "O people! I am only a human being and the Messenger of Allah. So I adjure you by Allah: if you know that I have fallen short in any way in conveying the messages of my Lord, then tell me, so that I may convey the messages of my Lord as they deserve to be conveyed; and if you know that I have conveyed the messages of my Lord, then tell me." The narrator said: So the people stood up and said: "We bear witness that you have indeed conveyed the messages of your Lord, have been sincere to your Ummah, and have fulfilled your responsibility." The narrator said: Then they fell silent. Then the Messenger of Allah (peace and blessings be upon him) said: "To proceed: some people claim that the eclipse of this sun and of this moon, and the departure of these stars from their places of rising, is because of the death of some great men among the people of the earth. They have indeed lied. Rather, these are signs among the signs of Allah by which He tests His servants, to see which of them turns in repentance. By Allah, since I stood up to pray I have seen everything that you are going to meet in your world and your Hereafter. By Allah, the Hour will not be established until thirty liars come out, the last of whom will be the one-eyed Dajjal, whose left eye will be wiped out as though it were the eye of Abu Tihya, an old man of the Ansar. When he comes out, he will claim that he is Allah. Whoever believes in him, affirms him and follows him, no previous good deed of his will benefit him; and whoever disbelieves in him and denies him will not be punished for any previous deed of his. He will prevail over the whole earth except the Haram and Bayt al-Maqdis, and he will besiege the believers in Bayt al-Maqdis, and they will be shaken with a severe shaking. Then Allah Almighty will defeat him and his armies, until the base of a wall, or the root of a tree, will call out: 'O believer! This is a disbeliever hiding behind me; come and kill him.'" He (peace and blessings be upon him) said: "That will not happen until you see matters whose gravity will weigh heavily in your hearts, and you will ask one another: 'Did your Prophet mention anything to you about these?' and until mountains move from their places. Then, after that, will come the taking (of souls)." And he (peace and blessings be upon him) pointed with his hand. The narrator said: Then I attended another sermon, and he said: He narrated this hadith, neither putting any part of it forward nor back.
This hadith is sahih (authentic) on the condition of the two Shaykhs (Bukhari and Muslim), but they did not record it.

Nafi' narrates from Sayyiduna Abdullah bin Umar (may Allah be pleased with them both) that on the day Sayyiduna Ibrahim (may Allah be pleased with him), the son of the Messenger of Allah (peace and blessings be upon him), passed away, the sun went into eclipse, and the people thought that the sun had gone into eclipse because of his death. So the Prophet (peace and blessings be upon him) stood up and said: "O people! The sun and the moon are only two of the signs of Allah; they do not go into eclipse because of anyone's death or life. So when you see that, stand up for prayer, remember Allah, supplicate and give charity."
This hadith is sahih (authentic) on the condition of Imam Muslim, but the two Shaykhs did not record it.