The Rambam enumerates three types of chillul Hashem. The first is when someone refuses to give up his life when called for. The second is when one commits a sin not because he’s driven by his urges but pretty much out of spite.
The third category is what we typically mean when we talk about chillul Hashem: when someone who should know better acts in a fashion that is perceived to be beneath him. In a Talmudic discussion of what would constitute a chillul Hashem, Rav said it would be if he didn’t pay his butcher on time (Yoma 86a). That’s not such a grievous thing but coming from Rav, it reflects badly on Torah scholars. Each of us at our own level is responsible to strive to act upward and not stoop down.
The reason for this mitzvah is gratitude to God. He created us and gave us everything. How horrible would we be to act in a fashion that makes Him look bad and causes people to say, “That’s how Jews act?” or “That’s how religious people act?” The sin of making a chillul Hashem is so serious that the Talmud in Yoma (86a) tells us that neither teshuvah (repentance), Yom Kippur nor suffering can fully effect atonement for a person. A person cannot be fully cleansed of the taint of making a chillul Hashem until he has died.
This mitzvah applies to both men and women in all times and places. It is discussed in the Talmud in the tractates of Yoma (86a) and Sanhedrin (74a-b), and is codified in the Shulchan Aruch in Yoreh De’ah 157. This mitzvah is #63 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #155 of the 194 negative mitzvos that can be observed today in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar