First of all, you must be 60+ to have an apprentice. Find someone below level 40 and right click his Avatar. Then, you'll get an option to send him an apprenticeship invite as shown below :
Ok, so what do you do next? Simple, just form a party with you and your apprentice ( You have to be party leader ) and speak to this npc ( The npc is the mastership/apprentice npc ) :
Ok so now you know how to get an apprentice. Let us move on to the mastership/apprentice quests.
Daily Quests :
Everyday, you get 2 daily quests. 1. Supported by the master ( Auto-quest. You don't have to do anything, it completes upon clicking )
2. Escort order (Complete cargo list with your apprentice in party. Once done, go back to changan and speak to the mastership/apprentice npc to receive your reward. (Reward is around 10 energy for the 1st quest, and I think it's 20 energy or 10 energy for the 2nd quest).
Special quests :
Special quests can be acquired by the apprentice every 5 lvl's. E.g 35-40, 40-45, 45-50 and so on.
You'll have to kill a legion boss or do a campaign. The reward will be 60-70 energy and morality points. The apprentice gets special gear for his lvl. The Morality points are only awarded to you if you do the campaign special quest, and daily cargo list (Thanks Samm for the edit). IMPORTANT EDIT : Just don't leave pt or switch realm once you take the apprentice quest. You'll fuck it up. Change realms/get your pt ready before you take the quest.
Morality are the points you accumulate every time your apprentice completes the special quests ( You also get more morality points when your apprentice graduates ). Here's an SS to show you where the mastership window is, and where morality points are displayed :
Once those morality points hit 800, your master title will be ranked up. With that title you can now buy a mastership seal from the mastership/apprentice npc ( kick-ass seals o.o )
Your apprentice graduates at 55. You get a high boost in morality points, and the apprentice gets some other stuff or whatever he gets ( idk my apprentice is still 50 D: ). Watch your affinity (It's in lineage screen). The more affinity you have, the more morality points you get upon graduation.
EDIT : Nah, you can't have more apprentices if you're lvl2 master. It COULD be at higher levels.
K, that's pretty much it. Others are welcome to correct/post new information about this system.