The Jedi quest was one experimental policy to train Padawans, used in the Almas Academy.
Jedi Master Nerra Ziveri, headmaster of the Academy since its inception in 119 BBY, had a considerable amount of Tarasin students from the nearby planet Cularin. Tarasins were culturally not prone to travel through the galaxy with a mentor, as was common with Padawans. Ziveri considered that exposure to other cultures too valuable for a Jedi, so he looked for alternatives. Firstly, he increased the courses on alien cultures.
But Ziveri also began the Jedi quests: During the training, a Padawan of the Almas Academy was supposed to have one quest every year from the moment she is twelve years old, till the moment she leaves the Academy, with a minimum of three at graduation. The quest always included the Padawan leaving the system, looking for something that the headmaster or the Padawan's mentor has asked for. The master sometimes went with the Padawan. Padawans older than fourteen who chose to remain in the Academy were to take a younger apprentice in their Jedi quests.
This initially included simple errands in Coruscant, Corellia and sometimes Kashyyyk (The Almas Academy included a good percentage of Wookiee students). When this method showed good results and the students showed beneficial changes, the masters began to meditate in the Force and test their students with more complex and long quests in the Core Worlds, the Corporate Sector and the Hutt Space.
Ziveri's successor as headmaster, Lanius Qel-Bertuk, continued this practice.