B”H is an abbreviation for Baruch Hashem which, in Hebrew, means “Blessed be the Name” (or Blessed be G_d).

It is also considered to be an abbreviation for BeEzrat Hashem, meaning “With the Help of the Name” (or With the Help of G_d).

Read from right to left, the Hebrew letters are bet and hey with ” symbolizing the abbreviated portion of the phrase. The Handbook of Jewish Thought by Aryeh Kaplan elaborates on the use of this abbreviation:

It is customary to write “bet-hey” … in the upper right hand corner of documents. However, some people prefer the abbreviation “bet-samech-daled” (BeSiyatta DeShamaya, Aramaic for “With the Help of Heaven”) in order to avoid writing any allusion to G_d’s name in a letter that might be thrown out. – Source: excerpted from The Handbook of Jewish Thought, Volume 1 by Rabbi Aryeh Kaplan.

The B”H abbreviation and the statement “Baruch Hashem” are closely tied to Orthodox branches of Judaism.