Definition: A Padawan, or Jedi apprentice, is a Jedi trainee who is apprenticed to a Jedi Knight or Master. Padawans receive one-on-one instruction in the ways of the Jedi. When a Padawan's training is completed, he must pass the Trials to become a Jedi Knight.
In-universe: In the early history of the Jedi, Jedi Masters could teach more than one apprentice at once. After the Jedi Order became more unified and centralized, around 4,000 BBY, the High Council set forth rules for training apprentices, who became known as Padawans. A Jedi Master could not take more than one Padawan at a time, and potential Padawans had to be under a certain age to be trained.
The rules and structure of Jedi training became even more strict and centralized after the Battle of Ruusan, around 1,000 BBY. The Jedi Order began to seek out infants with Force potential and raise them in the Jedi Temple, cut off from family and other emotional attachments. These Younglings were trained in the basic tenets of the Force, and had to pass the Initiate Trials in order to get chosen as Padawans. Some did not get chosen, and instead joined the Jedi Service Corps.
Padawans typically wore a single braid -- or equivalent jewelry, for species without hair -- to identify themselves as apprentices. They trained with their masters for about a decade, and were expected to obey their masters in everything as they learned and grew in the Force. If they passed the Jedi Trials, they became Jedi Knights. Generally, training an apprentice to Knighthood was a requirement for becoming a Jedi Master.
When Luke Skywalker reestablished the Jedi Order after the Jedi Purge, there were not enough fully-trained Jedi for the Master-Padawan system. Instead, Luke set up a Jedi Academy and trained many students, much like the Jedi of old. One-on-one apprenticeships existed, but were generally informal and impermanent. Later on, Luke's son Ben began to restore some of the Padawan tradition to the New Jedi Order, such as the Padawan braid.
- The Jedi Path: A Manual for Students of the Force by Daniel Wallace (2010)