A big part of the appeal of Downton Abbey lies in a fascination with the specific codes of manners and language that governed the lives of the landed gentry and their servants in England 100 years ago. Several scenes during the series depict how difficult it was to keep all the titles and honorifics straight: some American characters don't understand subtle distinctions; some nouveau-riche Britons use titles inappropriately; some noble characters invoke their titles to pull rank (or try to).
The dictionary definition of lady explains the complex ways the word is used in British high society, where it usually corresponds to the use of lord for men. For example, it's used when referring to women who hold certain titles: marchioness, countess, viscountess, or baroness. It can also be used of the wife of a lower-ranking noble, such as a baron, baronet, or knight.
Lady is also the courtesy title for the daughters of higher-ranking nobles: duke, marquess, or earl. (Earl is the British equivalent of count in European nobility). The daughters of viscounts and barons are referred to as "The Honorable," and daughters of baronets or knights are simply called "Miss."