Hadith: The Prophet (may Allah's peace and blessings be upon him) came out to us, and we said: O Messenger of Allah, we have learned how to greet you, but how should we invoke the blessings of Allah upon you?

عَنْ عَبْدِ الرَّحْمَنِ بْنِ أَبِي لَيْلَى قَالَ: لَقِيَنِي كَعْبُ بْنُ عُجْرَةَ، فَقَالَ: أَلاَ أُهْدِي لَكَ هَدِيَّةً؟
إِنَّ النَّبِيَّ صَلَّى اللهُ عَلَيْهِ وَسَلَّمَ خَرَجَ عَلَيْنَا، فَقُلْنَا: يَا رَسُولَ اللَّهِ، قَدْ عَلِمْنَا كَيْفَ نُسَلِّمُ عَلَيْكَ، فَكَيْفَ نُصَلِّي عَلَيْكَ؟ قَالَ: «فَقُولُوا: اللَّهُمَّ صَلِّ عَلَى مُحَمَّدٍ وَعَلَى آلِ مُحَمَّدٍ، كَمَا صَلَّيْتَ عَلَى آلِ إِبْرَاهِيمَ، إِنَّكَ حَمِيدٌ مَجِيدٌ، اللَّهُمَّ بَارِكْ عَلَى مُحَمَّدٍ وَعَلَى آلِ مُحَمَّدٍ، كَمَا بَارَكْتَ عَلَى آلِ إِبْرَاهِيمَ، إِنَّكَ حَمِيدٌ مَجِيدٌ».
[صحيح] - [متفق عليه] - [صحيح البخاري: 6357]

‘Abdur-Rahmān ibn Abi Layla reported: I came across Ka‘b ibn ‘Ujrah who said: Shall I offer you a present?
The Prophet (may Allah's peace and blessings be upon him) came out to us, and we said: O Messenger of Allah, we have learned how to greet you, but how should we invoke the blessings of Allah upon you? He said: "Say: Allāhumma salli ‘ala Muhammad wa ‘ala āli Muhammad kama sallayta ‘ala āli Ibrāhīm innaka hamīdun majīd. Allāhumma bārik' ala Muhammad wa' ala āli Muhammad kama bārakta' ala āli Ibrāhīm innaka hamīdun majīd (O Allah, bestow Your grace upon Muhammad and the family of Muhammad just as You bestowed Your grace upon the family of Abraham. Verily, You are Praiseworthy, All-Glorious. O Allah, bless Muhammad and the family of Muhammad just as You blessed the family of Abraham. Verily, You are Praiseworthy, All-Glorious)."
[Authentic hadith] - [Narrated by Bukhari & Muslim] - [Sahih Bukhari - 6357]

Explanation

The Companions asked the Prophet (may Allah's peace and blessings be upon him) how to invoke Allah's blessings upon him after learning how to greet him in At-Tahiyyāt: "As-salāmu ‘alayka ayyuha an-nabiyy wa rahmatullāhi wa barakātuh (O Prophet, may the peace, mercy, and blessings of Allah be upon you)..." The Prophet (may Allah's peace and blessings be upon him) informed them of the manner of invoking Allah's blessings upon him and its meaning: "O Allah, bestow Your grace upon Muhammad and the family of Muhammad," i.e., mention him, as well as those who follow his religion and the believers from among his relatives, with good praise among the highest assembly of angels. "Just as You bestowed Your grace upon the family of Abraham." As You bestowed Your favor upon the family of Abraham (peace be upon him), namely Abraham, Ismā‘īl (Ishmael), Is-hāq (Isaac), and their offspring and their followers from the believers, so bestow Your favor upon Muhammad (may Allah's peace and blessings be upon him). "Verily, You are Praiseworthy, All-Glorious," i.e., You are the Praiseworthy in Your essence, attributes, and acts; the All-Encompassing in Your greatness, sovereignty, and bounty. "O Allah, bless Muhammad and the family of Muhammad just as You blessed the family of Abraham," i.e., grant him the greatest of goodness and honor and increase it and establish it.