The Age of Discovery (AoD) expansion included the addition of tradeskill apprentices. In order to obtain your apprentice, you must be at least a level 10 crafter, and either own the AoD expansion or have purchased the tradeskill apprentice feature from the marketplace. Apprentices are located in North Qeynos inside the Ironforge Exchange, and in West Freeport on the patio of the Coalition of Tradesfolke building. (GU63 also added elite tradeskill apprentices for level 90+ crafters who own the AoD expansion, and those are covered here.)
It doesn't matter which apprentice you choose of the three (coldain male, human female, othmir) other than from an appearance perspective -- all of them research the same recipes, based on your crafting level and class.
Apprentices can be placed in any home for which you have trustee access, and multiple apprentices (from multiple alts) can be placed in the same home. You must, however, work only with the apprentice that your character is "master" of, so if you are going to keep the apprentices of all your alts in the same home, make sure they are named and/or dressed in a way that makes it easy for you to figure out who belongs to whom!
Once the apprentice is placed in a home, you first hail them and select which recipe you wish for them to research. Then you can pass along some of your knowledge to them (middle option on a hail) to reduce the recipe research time by 4 hours. You can then make a new piece of replacement equipment for them to reduce the research time by another 4 hours. You can do the coaching (pass on knowledge), and the crafting (replacement equipment), once per day, but be sure to coach before you hand in the crafted item, or you may not be able to coach that day due to a bug.
The items that you need to craft will always be tier-appropriate, and will cycle between the following (in the order listed):
All crafting classes get the ability to research the "greater essence" recipes at level 90, that create different appearance particle effects on reforged items.
For level 10 recipes, they are able to research anything in their subclass (scholar, outfitter or craftsman)
For level 20-90 recipes, the class breakdown is as follows:
If you have multiple alts to track, you might also find this downloadable apprentice recipe spreadsheet to be of use. This spreadsheet has now been updated to include the GU63 additions of elite tradeskill apprentices.