Within the SCA this person has a 'relationship' with a Laurel - who takes on the training or wellbeing of the Apprentice, and will promote and help them with their activities and crafts. An apprenticeship does not convey any official rank.
The nature of this relationship varies from Laurel to Laurel. Some Laurels require their apprentice to swear fealty. The duration of the apprenticeship is agreed upon beforehand by both parties, and could be until the apprentice becomes a Laurel, for a fixed time period or until the apprentice has advanced sufficiently to become a journeyman. Most apprentices wear some sort of token to indicate their status, often a green belt with the Laurel's device on it.
See Also: Lochac Laurels Guide to Apprentices
Apprenticeship can also be a level in a Guild. Some guilds replace "novice" with "apprentice."