To simplify, there are, generally speaking, three classes or ranks of Monsignors:
Papal Chamberlain, Honorary Prelate and Protonary Apostolic (P.A.)
Until 1968, the second rank was known as Domestic Prelate and such a Monsignor, along with the P.A., enjoyed a limited use of pontifical regalia and other perks otherwise reserved to bishops. One of these perks was the use of the title The Right Reverend Monsignor. The Papal Chamberlain was titled The Very Reverend Monsignor. Since 1969, all three classes are titled The Reverend Monsignor.
Today English-speaking Catholics associate the title The Rt. Rev. with abbots and* The Most Rev. *with bishops and archbishops, but as was mentioned above, prior to the inter-war era, The Rt. Rev. was associated with Catholic archbishops, bishops AND monsignors. Anglicans still use it for bishops, reserving *The Most Rev. *to their relatively small number of archbishops.
Further confusing the issue is the fact that in Italian, Spanish, Portuguese and French-speaking countries, Catholic bishops and archbishops are frequently addressed as Monsignor (which means My Lord), whereas in English-speaking countries we generally have used that terms only for priests who have been given that title by the Holy Father.