What do we mean when we say after the name of  Muhammad ,[peace and blessings of Allah be upon him]?

1- The meaning of blessings upon the Prophet (peace and blessings of Allaah be upon him) is a praise for him among the “higher group” (al-mala' al-'a'la, ie, the angels

Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have mercy on him) said:

The phrase “Bless Muhammad (salli ‘ala Muhammad)” – it was said that blessing from Allaah means mercy, blessings from the angels means prayer for forgiveness, and blessing from humans means du’aa’.

If it is said: “The angels sent blessings upon him,” it means that they prayed for forgiveness for him.

If it is said, “The khateeb sent blessings upon him,” it means that he prayed for blessing for him.

If it is said, “Allaah sent blessings upon him,” it means that He bestowed mercy upon him.

as Allaah says

انَّ اللَّهَ وَمَلَائِكَتَهُ يُصَلُّونَ عَلَى النَّبِيِّ ۚ يَا أَيُّهَا الَّذِينَ آمَنُوا صَلُّوا عَلَيْهِ وَسَلِّمُوا تَسْلِيمًا ﴿٥٦﴾
 (interpretation of the meaning👇): 

“O you who believe!send your Salaah on (ask Allaah to bless) him (Muhammad), and (you should) greet (salute) him with the Islamic way of greeting (salutation, ie As‑Salaamu 'Alaykum)” [al-Ahzaab 33:56].

2_There is also a prayer of the Prophet upon the believers:

وَصَلِّ عَلَيْهِمْ ۖ إِنَّ صَلَاتَكَ سَكَنٌ لَّهُمْ ۗ وَاللَّهُ سَمِيعٌ عَلِيمٌ (103)
(interpretation of the meaning👇): 

 and pray for blessings for them because your prayer will bring comfort to them; and Allah hears everything and knows everything.
(Al-Tawbah 103)

Praying here means:
Ask forgiveness for them, for their sins.