Slave Training is a re-education and reorientation process. Serving, pleasing and obeying are the foundations of slavery. A slave is trained and moulded to fit her Master's needs. A slave is no longer independent from her Master or his will. She is subject to whatever control he wishes to exert over her. In short, she is owned by her Master.
Slave training takes so many different forms, and can be handled in so many different ways that it would take a set of encyclopaedia to describe them all. You can train your slave to perform oral sex correctly, or train them to kneel in the right manner. You could train them to come when you play with their pussy, train them to dress to your preference, or train them to walk two steps behind you in public. Not all training needs to be private, and not all training needs to be painful.
Perhaps you like to train your slave girl by giving her a pat on the head and some nice words when she makes breakfast properly. Maybe you have a riding crop and deliver a quick whipping when your pony-girl doesn't raise her knees in a pretty trot. You are reinforcing behaviour, breaking her to your will, and moulding her to be your perfect companion.
Slave training is the ultimate in domination play. It runs the whole spectrum of BDSM play, and you can train your slave in any manner, to achieve any goal. The core of the exercise is not what you are teaching her to do: it is the fact that you are teaching her. Breaking her will, forcing her to submit, and ensuring your own dominance.