The Meaning of Salaat on the Prophet:
The Salaat of Allaah upon His Prophet has been interpreted to mean His praising of him before the angels, while the Salaat of the angels upon him has been interpreted to mean their supplication for him. This is the way Abul-'Aaliyah (radyAllaahu ‘anhu) has defined it as mentioned by Al-Bukhaaree in his Saheeh in the introduction of the chapter on "Verily, Allaah and His angels send their Salaat upon the Prophet. O you who believe, send your Salaat upon him as well as Salaam." [Surah Al-Ahzaab: 56]
Concerning the explanation of the angels' sending Salaat upon the Prophet, Al-Bukhaaree reported the following saying of Ibn 'Abbaas (radyAllaahu ‘anhu), after mentioning the (above) interpretation of Abul-'Aaliyah:
"To 'send their Salaat' means to send their blessings."
Meaning: They supplicate for him to be blessed.
The Salaat of Allaah on the Prophet (sallAllaahu ‘alayhi wa sallam) has also been explained as meaning His Forgiveness and Mercy on him. Al-Haafidh Ibn Hajr quoted many narrations (from the Salaf) in his Fat'h-ul-Baaree concerning this. He afterward commented on the Salaat being defined as such, saying:
"The most preferable and likely of these opinions (concerning the definition of Salaat) is that which was mentioned previously from Abul-'Aaliyah that the meaning of Allaah's Salaat on His Prophet is His praising and honoring of him, while the Salaat of the angels, as well as everyone else, upon him means their requesting the same thing (i.e. praise and honor) for him from Allaah. However, what is intended here by their request is that of an increase in praise and honor and not the same amount as intended by the original Salaat."
Al-Haafidh (Ibn Hajr) said: "Al-Haleemee said in ash-Shu'ab:
'The meaning of Allaah's Salaat on the Prophet is His extolling of him.'
So what is meant by our statement: 'Allaahumma Salli 'alaa Muhammad' is 'O Allaah, Exalt Muhammad.' What is intended by this is: Extolling him in the worldly life with the highest form of remembrance, while manifesting his Religion and preserving his legislation. As regards to the Hereafter, what is intended by it is that he (sallAllaahu ‘alayhi wa sallam) be given tremendous reward, granted the ability to intercede on behalf of his ummah and that his virtue be manifested in the praiseworthy station.'
So based on this, what is meant by Allaah's statement "Send your Salaat upon him" is: "Invoke your Lord to send His Salaat on him."
The great scholar, Ibn Al-Qayyim said in his book "Jalaa-ul-Afhaam fis-Salaati was-Salaam 'alaa Khair-il-Anaam" while discussing the meaning of the Salaat of Allaah and His Angels on His Messenger (sallAllaahu ‘alayhi wa sallam) and His commanding the believing servants to send their Salaat on him, after refuting the claim that the meaning of Salaat is Mercy and Forgiveness:
"Rather, the Salaat that the people were commanded with in it - i.e. the ayah in Surah Al-Ahzaab - is a request to Allaah that he implements what He has informed us about concerning His Salaat and the Salaat of the Angels, and that is: Praising him, manifesting his virtue and nobility, and seeking to honor and elevate him." So it consists of a notification and a request. This asking and supplicating on our part was called "Salaat on him" for two reasons:
First: It entails that when a person sends Salaat on him, he is praising him and extolling the mention of his virtue and nobility, as well as desiring and loving for Allaah that He do the same. So it consists of a notification (i.e. of a person's praises) as well as a request (from Allaah that He send His praises).
Second: Such an action was called Salaat from us due to the fact that we are asking Allaah to send His Salaat on him (i.e. the Prophet). So therefore, the meaning of Allaah's Salaat on the Prophet is His praising of him in order to elevate his remembrance and rank with Allaah, while the meaning of our Salaat on the Prophet is: Asking Allaah to do that (i.e. the above) for him." [End of Ibn Al-Qayyim's words]
The Meaning of Salaam on the Prophet:
As for the meaning of Salaam on the Prophet (sallAllaahu ‘alayhi wa sallam), Al-Majd Al-Fayrooz Abaadee said in his book As-Silaatu wal-Bushr fis-Salaati 'alaa Khair-il-Bashr:
"It means: As-Salaam - which is one of Allaah's Names - be upon you. And the interpretation of that is: 'May you not be void of goodness and blessing, and may you be safe from detestable and harmful things.' This is since a Name of Allaah is only mentioned upon certain things in the hopes of gathering all of the meanings of good and blessing for those things, and to remove the occurrence of defects and corruption from those things.
It is befitting that the Salaam (here) take on the meaning of Salaamah (i.e. peace/safety). Meaning: May Allaah's divine decree on you be that of peace and security - i.e. May you be safe and secure from blemishes and defects.
So if you say: 'O Allaah, send your Salaam on Muhammad', then what you intend from this is: 'O Allaah, decree for Muhammad - with regard to his Call, ummah and remembrance - safety and absolvement from every defect. Thus throughout the passage of time, let his Call increase in status, his ummah in numbers, and his remembrance elevation.'"