This is a person's first stage in their Jedi training. Jedi Hopefuls are very young children taken to the Jedi Temple to be trained under Jedi Masters such as Yoda. These Jedi Hopeful will usually be trained in a group of pupils and will live at the Jedi Temple. Because they are taken at such young ages, most have never known their true parents; thus they have few problems coping with the "strange" environment of the Jedi Temple. Jedi Hopefuls remain training at the Jedi Temple until they are chosen to be a Padawan learner by either a Jedi Knight or Master. If by a certain age a Jedi Hopeful has not been selected to become a Padawan, they will be sent to work in community service of the Galactic Republic. Depending on their skills, they may be supervised to ensure that they do not cause trouble through their Jedi knowledge.
Becoming a Jedi Hopeful is a great honor. They are usually choosen by Jedi Masters and Knights who encounter them on their journeys throughout the galaxy.
Anakin Skywalker was accepted into the Jedi Order at an older age, thus he skipped the Jedi Hopeful stage of his training and became a Padawan learner of Jedi Knight Obi-Wan-Kenobi.