What does B"H mean?

B"H (or BH) is an acronym for the Hebrew words "Baruch Hashem," which means, "Blessed be G‑d." Traditionally, Jewish people begin letters and correspondences with these letters, in part to contextualize what's most important, and in part to remind us that all comes from Him—including the contents of the communication to follow.

With regards to emails, I know very few who are careful to do the same with email, probably since it's used so frequently and informally. It's not an iron-clad requirement for letters either, rather a tradition that people have accepted upon themselves.