I am aware that Mosafeheh (المصافحه) is a kind of greetings which is famous between Muslims as an Islamic deed and apparently has a Thawab (reward). Actually it seems to be shake-hands. But I'd like to know regarding the circumstances of Musafehe: how precisely shall we do it?
المصافحة is merely the action of shaking hands, and it is not specific to Muslim.
It is mentioned that if a muslim were to greet another muslim, and shake hands, his sins would fall away like the leaves of a tree.
إِنَّ الْمُؤْمِنَ إِذَا لَقِيَ الْمُؤْمِنَ فَسَلَّمَ عَلَيْهِ وَأَخَذَ بِيَدِهِ فَصَافَحَهُ تَنَاثَرَتْ خَطَايَاهُمَا كَمَا يَتَنَاثَرُ وَرَقُ الشَّجَرِ
when a believer meets another believer and gives greeting and takes his hand and shakes it, their sins/mistakes would fall away like the leaves of a tree fall away.