Apprentice is the first rank in the Dark Brotherhood. It is given to members who have passed the Test of Lore and gotten placed in a House. Apprentice is also a more general term that refers to any member under the rank of Dark Jedi Knight who has been taken for training by a sanctioned Master in the Master-Student Program.
Most often when you hear someone refer to an Apprentice they are speaking of the Master-Student Program conotation of the term.