"Battle Advantage" is the amount of extra troops you can field at a time during a battle. This is calculated using several factors, such as the actual sizes of the sides of the battles, the tactics rating of the 'leaders' of each side of a battle, and the 'battle size' you have selected in your options. For example, if a Scottish army of 200 men is facing an English army of 1000 men, it can be expected that the English side will have a significant battle advantage.
Here is an example to illustrate:
Let us assume you have set the size of battles in your options to 70. That means that roughly 70 men will be present at any given time during a battle scene. If you engage in a battle in which both sides have nearly equal amounts of troops and the leadership values of the respective generals are equal, then each side will have a battle advantage of '0'. This means that when men spawn on the battefield, each side will start with 35 men (equal amounts). If, on the other hand, you have a larger force or you have a higher tactics rating that makes your battle advantage '2', for example, you will be able to field 2 more men than usual (37 men) and your opponent will consequently have 2 less men to keep the battle size of 70 the same (33 men). Obviously, 37 men will have an easier time of winning against 33 men, hence you have a 'battle advantage'.
It is generally unwise to engage in a battle where you would have a large battle DISadvantage (battle advantage is a high negative number like -10) as even elite troops will be swarmed in such a situation, making victory unlikely. It should also be noted that, in general, the amount of renown you gain is affected by battle advantages and you will gain more renown for winning a battle when you are at a disadvantage.