The Champion Ability system - also being called sub-classing, to my dismay - is your reward for completing Champion Levels. The idea is to add utility to classes by letting them get weaker versions of the abilities of other base class types. If you're a fighter, for example, you can get a direct damage spell or a heal.

Champion Abilities are organized first by base class (Acolyte, Magician, Viking, etc) and then into paths (a.k.a. lines). You cannot train in your own base class' paths - if you started as a fighter, the fighter trainer won't train you (Vampiirs are considered Guardians). Each path has a focus - Cold-based damage spells, group heals, weapon styles, and so on. Training in a path opens up the next level - if you don't have one point in a particular path already, you can't get the second ability in that path.

To get A3, for example, you must have both A1 and A2. For the lines that merge, you can get either C1 and C2 or D1 and D2 in order to get abilities 3, 4, and 5.

You can mix and match your points as you wish - you can get four abilities in one path and one in another, or three from one path and two from a path on a different class, or any combination you want; but since there are only five (or ten with the Labyrinth expansion) Champion Levels, you get only five (or ten) Champion Abilities.

You can respec your Champion Abilities by purchasing a Cambriare stone from the Quartermaster in the Throne Room instance - if you are at least CL3 - and handing it to the Weaponmaster. If you want to play around with other CL Abilities before you reach CL3, you can copy to Pendragon - the stones are available for purchase in the capital cities at the same imp that sells other respec stones.