What is the meaning of "O Allah, send prayers upon our Master Muhammad and upon the Family of Muhammad"? And is this statement specific to the Family of Muhammad, or is it permissible to say "O Allah, send prayers upon so-and-so" for any person?

Prayer from Allah Almighty means mercy and mention in the Highest Assembly. The Household of Muhammad, may Allah's peace and blessings be upon him, are specially designated for mention. This refers to the Prophet's wives, may Allah's peace and blessings be upon him, and the believers from Banu Hashim and Abd al-Muttalib. As for other believers, they are included in the worshipper's statement: "Peace be upon us and upon the righteous servants of Allah."