What does it mean when we say "Allahumma Salli ^Ala Muhammad Wa Sallim"? and why do people write "SAW" after the Prophet's name?
It means: "O Allah raise the rank of Prophet Muhammad, and protect his nation from which he fears for it".
The one who says the salat onthe Prophet is promised with great benefits and reward.
قال تعالى: ( ان الله وملائكته يصلون على النبي يا أيها الذين ءامنوا صلوا عليه وسلموا تسليما) سورة الأحزاب ءاية
It means Allah has raised the rank of His Prophet (Muhammad) and honored Him. The Angels do also pray and ask Allah to elevate Prophet Muhammad's rank and honor. O believers ask Allah to raise the rank of the Prophet and to Protect his nation.
قال رسول الله صلى الله عليه وسلم: "من صلى عليّ يوم الجمعة ثمانين مرة غفرت له ذنوب ثمانين عاما"
Our Prophet said that: "Whoever says Salat on me in a Friday 80 Times, Allah will Forgive for him 80 Years of Sins".
He also said:
قال رسول الله صلى الله عليه و سلم "من صلى عليّ في كل يوم مائة مرة قضى الله له مائة حاجة سبعين منها لآخرته و ثلاثين منها لدنياه.ـ
Whcih means: "Whoever says Salat on me everyday one hundred times, Allah will fulfil for him one hundred needs, seventy of them for his hereafter (Akhirah) and thirty of them for this life (dunya).
Note1: We say say "Allahum Salli" صلِ with Kasra, not ya' since Ya' is feminine.
Note2: It's disliked to write "SAW" after the name of the Prophet. To get reward, we should write "Sallallahu ^Alayhi Wa Sallam"