See also the Tournament-specific version, which has additional clarifications and examples.
Terrain Modifier: Bonus conferred upon an individual army in a specific terrain, for example Spiders get a +2 Terrain Modifier within a city. These bonuses cannot be cancelled.
Hero Bonus: Bonus that applies to the entire stack due to the natural command bonus of the strongest hero in the stack, plus the total of any Command Items carried into battle. The natural command bonus depends on the Hero Strength. These bonuses are cancelled by an army with the Cancel Hero Attribute. If there is more than one hero in a stack, the effective hero bonus is the highest natural (strength-based) hero bonus plus the total of all command items. (This was verified with actual battles - the View Stack command gives incorrect information in this case). Naval heroes do not receive bonuses, but can confer them on non-naval units.
City Bonus: Bonus conferred upon the defending stack for being within a City, Tower or Special Location. It is +2 for temples, ruins, and player-owned cities that are capable of producing 3 or more armies. It is +1 for towers, neutral cities capable of producing 3 or 4 units, and player-owned cities that are capable of producing 2 or less armies (including cities that don't produce anything). It is zero for razed cities and neutral cities capable of producing 0, 1, or 2 units. It is cancelled by an army with the Cancel City Attribute.
Fortify Bonus: Bonus conferred upon a stack by an army with the Fortify Attribute, regardless of terrain, and regardless of whether the stack is attacking or defending. (Tested in actual battles.) Typically it is +1 (the Scenario Builder will not allow +2 or +3...) There are no examples of this in the default (Erythea) army set. This bonus cannot be cancelled. Naval units do not receive any bonuses, but can confer them on non-naval units.
Non-Hero Bonus: Bonus that applies to the entire stack, but which is not due to a Hero, City, or Fortify bonus. For example Dragons give a +2 Non-Hero Bonus everywhere while Wolfriders confer a +1 Non-Hero Bonus only in the Hills. Only the highest single Non-Hero Bonus will be applied to the stack. These bonuses are cancelled by an army with the Cancel Non-Hero Attribute. Naval units do not receive any bonuses, but can confer them on non-naval units.
Total Bonus: Sum of all non-cancelled Non-Hero, Hero, City and Fortify Bonuses. May be no larger than 5.
Enemy Subtract Modifier: Negative bonus that is subtracted from an enemy's stack after all other Bonuses are accounted for. Only the highest single Modifier will be subtracted. It cannot be cancelled. Naval units cannot receive an Enemy Subtract Modifier, but may confer it upon non-naval (e.g., flying) units in the enemy stack.
Cancel Non-Hero Attribute: Attribute of an army (typically an ally) that cancels the Non-Hero Bonus. If you have an army with the Cancel Non-Hero Attribute, any enemy Non-Hero Bonuses will not be applied. This does not cancel any Terrain Modifiers nor the Enemy Subtract Modifier. A naval unit with a Cancel Non-Hero Attribute will still cancel the enemy's Non-Hero Bonuses!
Cancel Hero Attribute: Attribute of an army (typically an ally) that cancels the Hero Bonus. If you have an army with the Cancel Hero Attribute and a Hero is present in the enemy's stack, he will not confer his natural command bonus nor bonuses from any command items to any armies in his stack, including himself. A naval unit with a Cancel Hero Attribute will still cancel the enemy's Hero Bonuses!
Cancel City Attribute: Attribute of an army (e.g., a Catapult) that cancels the City Bonus. If you have an army with the Cancel City Attribute and you are sieging a city, tower or Special Location, the enemy will not receive any bonus conferred from being inside the structure's walls. This does not cancel any individual city Terrain Modifiers, Non-Hero or Hero Bonuses.
Base Strength: Strength of an army before taking any of the above Bonuses or Modifiers into account but including blessings and medals. May be no larger than 9.
Hero Strength: Base Strength of a Hero, plus any Battle Items, but not including any of the above Bonuses, including those due to Command Items. May be no larger than 9. The Hero Strength determines the hero's natural command bonus as follows:
Terrain Strength: Strength of an army in the applicable terrain (Base Strength plus Terrain Modifier, if any). May be no larger than 9. All battles take place in the terrain of the defender.
Modified Strength: Terrain Strength plus Total Bonus. May be no larger than 14.
Final Strength: Modified Strength minus any Enemy Subtract Modifiers. May be no smaller than 1.
Special Location: A temple or ruin (including strongholds). Treated as terrain type city and confers a +2 City Bonus. This bonus can be cancelled by an army with the Cancel City Attribute.
Tower: A protective structure resulting from a stack being given the "defend" order (in the Orders Menu) at least one turn ago. Treated as terrain type city (regardless of actual terrain) and confers a +1 City Bonus. This bonus can be cancelled by an army with the Cancel City Attribute. Towers cannot be built in Special Locations, cities or in terrain types water, shore or mountains. A ruin with "Tower" in the name is not a Tower.
Medal: A permanent +1 modifier to an army's Base Strength occasionally awarded to a non-ally army accompanying a Hero in a battle against another player. Armies may receive more than one medal.
Blessing: A permanent +1 modifier to an army's Base Strength resulting from a unit doing a search within a temple. Armies may only be blessed once by each temple, but blessings from multiple temples are cumulative. An army's Base Strength can never increase above 9.