Which of these is a kind of letter sent out by the Pope?
Although the term "encyclical" originally simply meant a circulating letter, it acquired a more specific meaning within the context of the Catholic Church. In 1740, Pope Benedict XIV wrote a letter titled Ubi primum, which is generally regarded as the first encyclical in a modern sense. The term is now used almost exclusively for a kind of letter sent out by the Pope.
For the modern Roman Catholic Church, a papal encyclical is a specific category of papal document, a kind of letter concerning Catholic doctrine, sent by the Pope and usually addressed especially to patriarchs, primates, archbishops and bishops who are in communion with the Holy See. The form of the address can vary widely and may concern bishops in a particular area, or designate a wider audience. Papal encyclicals usually take the form of a papal brief due to their more personal nature as opposed to the formal papal bull. They are usually written in Latin and, like all papal documents, the title of the encyclical is usually taken from its first few words (its incipit).