The concept of Vigor within the context of battles represents the fact that early on in a battle, a character is much more energetic and capable of performing special maneuvers and/or attacks. As the battle wears on the character slowly loses steam and eventually defaults to a basic hack and slash style of fighting. To represent this Vigor is tracked as a resource, called Vigor points, or simply Vigor.
Vigor points are an expendable resource, with a maximum value set as a function of Strength. As characters level up, their Vigor max increases. At any given level the Vigor max can be calculated as follows:
VIGOR MAX @ lvl1-10 = STRENGTH x LEVEL/2
VIGOR MAX @ lvl11-20 = STRENGTH x [LEVEL/4+2.5] (Full class gets +1/level on top of this formula)
NOTE: see your Class Codex for a simplified calculation per level for your specific Class
Each Vigor skill has a fixed Vigor cost. As Vigor skills are used the appropriate amount of Vigor must be removed. If you have less Vigor than the skill you wish to attempt, it cannot be used. If there is sufficient time between battles (i.e. the battles are not back-to-back) then Vigor is reset to its maximum amount prior to a new battle beginning.
For prolonged combat encounters (i.e. dungeon crawls the retain movement and Turn Order restrictions) assume that after three rounds of no new combat-oriented instants or actions being used, Vigor resets to maximum.
Vigor classes gain two new skills each level, up to level 10. After level 10 they gain only one new skill each level. When choosing new skills, they must be equal to or less than current level of the character. As most new characters start at level 1, any new Vigor class character would select two level 1 skills to start with. This is the only time under normal circumstances that learning new skills does not require training.
Vigor classes cannot learn a previously unknown skill (i.e. Offensive Stance) unless they are taught by a Combat Trainer who knows the corresponding skill. If the Vigor class has lv3 of the Combat Trainer Secondary Skill they can self-teach new skills. If upgrading an already known skill (which is any skill with a pre-requisite listed) then no trainer is required and the skill can be learned immediately (i.e. upgrading Offensive Stance into Improved Offensive Stance). This represents a Vigor class honing their skills rather than actually learning a new skill
Eligible combat trainers for Vigor classes can be found in most major settlements, though they are less common in Wild Elf settlements. The going rate for learning a new Vigor skill is 5 gold; however, this cost is usually waived if the skill is taught by a guild or other such organization (i.e. a senior officer training a soldier). In some role-playing specific instances it may also be reasonable to assume that an skill is taught at no cost (i.e. a father teaching his son or a an elderly sword master passing on a skill to his young protégé).
For a complete list of Vigor skills available to learn see the Vigor Skills page.