Complex sentence: When a person (traditionally the wife in many cultures) assumes the family name of his or her spouse, that name replaces the person's birth name, which in the case of the wife is sometimes called the maiden name.
Split into simpler sentences: When a person (in many cultures, traditionally, the wife) assumes the family name of his or her spouse. That name replaces the person's birth name, which in the case of the wife is sometimes called the maiden name.