A department of the Roman Curia.
The dicasteries are the Secretariat of State, Congregations, Tribunals, Councils, and Offices.
Other bodies are not dicasteries, but simply institutes of the Roman Curia.
EWTN provides the technical details for Contacting a Roman Dicastery. Outsiders often imagine that the dicasteries are large, sleek offices filled with the latest technology and the great numbers of people needed to operate an institution with charge over more than 1 billion Catholics worldwide. Anyone who has ever visited one of these offices will swiftly be better informed. The number of persons working in each dicastery is far smaller than what we’d expect, and the equipment they use is often older.
Catholics who decide to write to the Vatican often discover that nothing happens after they write a letter. Most often the dicastery refers their letter back to the local bishop with a request that he take care of the matter. Let me comfort you. Even priests who write to a dicastery often get the same result.
The key is remembering that the dicasteries are sized and staffed to handle correspondence from the world’s bishops. When Rabbi Yeshua walked the earth, people didn’t usually go straight to him. They went to a shaliakh, who would, or would not, introduce them to Rabbi Yeshua. Bishops are successors of the shlikhim, and we too go to them to get to the Vicar of Christ.
It makes sense. Sometimes a dicastery will get a letter from a lay Catholic, “My priest isn’t following the rubrics.” First, the bishop, not the Vatican, is the father of his diocese. The priest is his responsibility. The bishop knows his priests well, and can quickly assess whether an issue raised by a parishioner is valid or not.
The bishop reports directly to the Vicar of Christ. Every five years each diocesan bishop travels to Rome for his five-minute ad limina personal visit with the pope. The pope is always well-briefed for these visits by the dicasteries, and he asks each bishop direct questions about what is happening in his diocese. It’s surprising how much the pope can get done in five minutes!
Certainly the pope expects his bishops and dicasteries to work together. But, in the end, the Curia is there to serve the pope’s needs.