Hadith: Here is Jibrīl giving you greetings of peace

عن عائشة رضي الله عنها قالت: قال لي رسول الله صلى الله عليه وسلم : «هذا جِبْرِيلُ يَقْرَأُ عليك السَّلام» قالت: قلت: وعليه السلام ورحمة الله وبركاته. وهكذا وقع في بعض روايات الصحيحين: «وبركاته» وفي بعضها بحذفها، وزيادة الثقة مقبولة.
[صحيح] - [متفق عليه]

‘Ā’ishah (may Allah be pleased with her) reported that the Prophet (may Allah's peace and blessings be upon him) once said to her: “O ‘Ā’ishah, here is Jibrīl giving you greetings of peace.” She said: “And on him be peace, mercy, and blessings of Allah.” Some narrations in Sahīh Al-Bukhāri and Sahīh Muslim included the words “and blessings” while others did not include them. The addition made by trustworthy narrators is accepted.
[Authentic hadith] - [Narrated by Bukhari & Muslim]

Explanation

‘Ā’ishah (may Allah be pleased with her) informs us that the Prophet (may Allah's peace and blessings be upon him) told her: “O ‘Ā’ishah, here is Jibrīl giving you greetings of peace.” i.e. he is greeting you with the greeting of Islam. So she said: “and on him be peace, mercy, and blessings of Allah.” Hence, she returned the greeting with a greeting better than it, in compliance with the divine command: {And when you are greeted with a greeting, greet in return with one better than it or [at least] return it [in a like manner]} [Sūrat An-Nisā’: 86]. It is Sunnah that if someone conveyed greetings of peace from one person to another that the latter answers saying: “and on him be peace, mercy, and blessings of Allah.” This is the apparent indication of the above Hadīth by ‘Ā’ishah (may Allah be pleased with her). If the person says: “On you and on him be peace” or “On him and on you be peace, mercy, and blessings of Allah,” then that is also good because the one who relays greetings of peace is doing a righteous deed, therefore you should reward him by supplicating Allah to give him peace, mercy, and blessings. Are you obliged to relay the greetings if someone says: “Send my greetings of peace to so and so”? Scholars elaborated on this matter saying if you verbally commit yourself to do that, then it becomes an obligation on you, because Allah says: {Indeed, Allah commands you to render trusts to whom they are due} [Sūrat An-Nisā’: 58]. However, if someone says: “Send my greetings of peace to so and so” and you remain silent or you said: for instance, “I will if I remember” or something similar, then in this case you are not obliged to relay the greetings except if you remember to do so. Nevertheless, it is better that a person does not burden another with this task as it might be difficult for him. He should, rather, say: “Send my greetings of peace to whoever asks about me.” This is a good thing to do, whereas burdening him with this task is not appropriate because he might feel embarrassed and say: “Yes, I will send your greetings,” but then he might forget, or a long time might elapse before he fulfills this task.