Jedi Master was the second highest rank a Jedi could obtain, and a traditional requirement for becoming part of the Jedi Council, the main interface between the Jedi and the government of the Republic. It was reserved for those who have shown exceptional devotion and skill in the Force. The titles of Sith Lord or Sith Master were on par with Jedi Master.
The rank was usually achieved by Knights who successfully entered Padawans into knighthood. When a teacher felt that his or her Padawan was ready, the Council would usually promote the Padawan to Jedi Knight and the teacher to Jedi Master under the proviso that the pupil pass a series of tests, known as the "trials". The Master could then pick another apprentice. A variation of this practice occurred when Jedi were promoted to Master having served as an instructor at a Jedi Academy or Praxeum, training many students instead of one individual Padawan. Dav Man'Sell of the Jedi Praxeum of Yavin IV was promoted to Jedi Master at least partially under this provision, having served as a chief Lightsaber Instructor at the Praxeum. Dav Man'Sell later trained Padawan Joshua Kierra-Solo to Knighthood.
A Knight who hadn't taken a Padawan could also retake the trials to earn the rank; often the trials were much harder and often involved a few more tests. The rank could also be granted by the Jedi Council to a Knight for performing some extraordinary deed or service as in the case of Steffo Rancis, although this was rare and often reserved for Padawans to become Knights. Even rarer, and often frowned on, was for a Knight to assume the rank without the Jedi High Council's approval. There were few who were known to have done this, and the majority earned the rank officially, rather than deeming it upon themselves.
Jedi Masters were seen as knowledgable and resourceful, and having a high level degree of understanding in the force and its applciations. Most were fondly regaurded and deeply respected for their power, insight and knowledge. They were looked to in times of peril and great need for comfort, support and guidence.
Jedi Masters were known for their inner calmness as well as the calmness they exuded, they were often very experienced in many different aptitudes and skills. They led from the front and could be capable of dramtic action when necessary. They represented all that was good, and stood as beacons of light and hope for their fellow Jedi, the Jedi Order, and the entire galaxy.
The wisest of Jedi Masters sometimes went on to serve on the Jedi High Council. Many served on localised Master's councils, which often took responsibility for the running of individual Temples or Praxeums, particularly during times when there was not a universally recognised High Council. The Corellian Jedi Order, the Jedi Praxeum of Yavin IV, and the Osarian Jedi Order were three such worlds that at various points had their own council, though most of these devolved councils later opted to defer to the leadership of the High Council once it was re-established.