What does it mean to desecrate or sanctify God’s name? A fascinating array of answers has been given, each of which explains the above verse in a different way.
The classical interpretation is that the verse speaks of extreme situations. Rashi explains it as relating to martyrdom – a directive for Jews to give up their life for the sake of their religion if forced to transgress it. The phrase “among the children of Israel” denotes this is an act of public significance at a time of persecution, not of an individual, but of the people as a whole.

The Talmud, however, has a very different take: the verse does not relate to a one-off extreme situation but rather to the everyday conduct of those who study and teach Torah and are associated with God’s name (Yoma 86a). When they conduct themselves pleasantly and with respect for others, then God’s name is sanctified. When they conduct themselves with arrogance and are untrustworthy, they desecrate His name. 

In a secularised but related fashion, Israelis leaving the country at Ben Gurion airport are confronted with a sign, in Hebrew, which reads: “Overseas: You Are the state”. 
Sanctification of God’s name is thus the requirement to ensure that Jewish religion, values and identity are not brought into disrepute through the behaviour of those who represent them but rather are elevated through their conduct.

A final interpretation, which  places the verse in its wider context, is that of Ibn Ezra. As Parashat Emor begins with Moses addressing the priests, so this section concludes with this commandment, which applies uniquely to them, the priests who must sanctify God’s name “among the children of Israel”.