Affection is a stat that determines how much a girl likes you and their relationship level. There are currently a total of 8 relationship levels.
Once you completely fill the affection bar, the girl will reach the next relationship level. To get a girl to Girlfriend level and beyond, there are additional requirements that must be met. These requirements can be viewed by clicking on the girl's portrait in-game. They are also listed under each girl's page on this wiki.
Every action that recovers a girl's stats will raise her affection by some amount, as long as they have the energy and the stat is not at its max. Affection gain is reduced if the stats recovered by the action is greater than the max.
If you allow any of the girls' stats to reach zero, they will be forced to take care of themselves. This resets the stat to 50 at the cost of losing some affection. The affection lost increases each consecutive time a girl has to take care of herself, so make sure you don't neglect your girls.
There are checkpoints at 12.5%, 25%, 37.5%, 50%, 62.5%, and 75% affection. Once you reach these, affection will not drop below these values. It is also impossible to drop relationship levels, so you don't have to worry about that either. There are no checkpoints for girls who have reached their highest available level, since there is no further content beyond that level.