Basic Information
A wizard's apprentice is a person who is serving an apprenticeship to a wizard.
This will generally imply that they are not attending a wizarding school unless the title is an honorific for someone attending such.
The apprenticeship may be arranged under the auspices of a formal guild or may be a less regulated one-to-one arrangement. Depending on the character of the master wizard the apprentice may be bonded labour, an experimental animal or an actual protegee - a guild is likely to abate the worst excesses to be found in freeform apprenticeships but may not entirely remove them.
In some settings, an apprenticeship may be compulsory for anyone with "the gift" - either you are properly trained to use your magic or it gets suppressed somehow (this especially in settings where magic is dangerous).