Koukan literally means good feelings or good intentions. So koukan points indicate how the characters feel towards you. The points aren't displayed as numbers though. Instead, the koukan points are shown as a ranking list like the example below.
In the above example, this was a screenshot taken before starting Chapter 1 for the first time, which means all the characters are neutral and have equal koukan. To determine a character's koukan level, just look at their face. The happier they look, the higher their koukan points.
So what does koukan points have to do with the game? Koukan points will determine which epilogue picture you will see at the end of the game (if at all). For example, if Inuyasha hs the most koukan points, then you will see Sengoku Emaki 25. However, if none of the characters like you that much then you won't see any epilogue pictures.
Increasing Koukan Points
There are a couple of things that can make a character's koukan go up or down. The most common way to increase a character's koukan is to use the character in the chapters, win battles, and finish the chapter quickly. Likewise, a common reason for a decrease in koukan is if you don't use the character, or if you do use the character in a chapter, you lose battles and take ages to finish the chapter.
Another way to increase/decrease koukan is white suqare events. When you do a chapter, you'll notice different coloured squares: red, blue, green, yellow and white. The white squares trigger events that promts you to select an option that will affect the koukan of the character you are using. Although sometimes nothing will happen if you land on the square.
One common event is the 'take a break' event. When you land on a white square the character will ask you if you are tired and want to stop for a rest. You are given two options: 1) I'm fine, and 2) Let's stop for a while. In most cases, you should choose "Let's stop for a while" if you want to increase koukan.