Teaching the doctrines and principles of the gospel is a primary responsibility of the Relief Society presidency.
The Melchizedek Priesthood and the Relief Society study the same lessons, except for the first Sunday of each month. This parallel schedule facilitates gospel study and discussion in the home.
However, Relief Society sisters and Melchizedek Priesthood brethren meet separately, even on weeks when the same lesson is being studied. An exception may be made for fifth-Sunday discussions, which may be conducted under the direction of the bishopric in a meeting that combines the brethren and sisters.
In addition to information provided about Sunday instruction here, see "Relief Society," section 3 of the Church Handbook of Instructions, Book 2: Priesthood and Auxiliary Leaders (1998), 201.