The Three Branches of the Gwageo
The Joseon gwageo had three branches: mun-gwa, mu-gwa, and jap-gwa. Mun-gwa selected civil officials, mu-gwa selected military officials, and jap-gwa selected specialists in fields such as medicine, astronomy, and interpretation.
Source: 위키백과 조선의 과거제 (Wikipedia, Korean: "Joseon-era gwageo system")
Sogwa and Daegwa
Sogwa refers jointly to the saengwonsi and jinsasi exams; passing one of them was required to sit the daegwa (the mun-gwa exam itself). Daegwa refers to the mun-gwa exam proper.
Source: 위키백과 조선의 과거제 (Wikipedia, Korean: "Joseon-era gwageo system")
The Three Stages of Mun-gwa
Mun-gwa was held in three stages, in order: chosi, boksi, and jeonsi. Chosi selected candidates in proportion to each province's population, boksi narrowed the field to 33 candidates, and jeonsi, held before the king, decided their final ranking.
Source: 우리역사넷(국사편찬위원회) 과거 제도 (Woori History Net, National Institute of Korean History)
Siknyeonsi and Irregular Exams
The regular exam, siknyeonsi, was held every three years. Irregular exams were also held: jeunggwangsi for national celebrations, byeolsi for special occasions, and alseongsi when the king visited Seonggyungwan in person.
Source: 위키백과 조선의 과거제 (Wikipedia, Korean: "Joseon-era gwageo system")
Mu-gwa
Mu-gwa was a single track with no sogwa/daegwa split, but like mun-gwa it had three stages: chosi, boksi, and jeonsi.
Source: 위키백과 조선의 과거제 (Wikipedia, Korean: "Joseon-era gwageo system")
Jap-gwa
Jap-gwa selected technical specialists: uigwa (medicine), eumyanggwa (astronomy and geography), yeokgwa (interpretation), igwa (administration), and others.
Source: 위키백과 조선의 과거제 (Wikipedia, Korean: "Joseon-era gwageo system")