In many of these stories, the regular girl has some unique qualities. Maybe she's really good at art or has a great love for animals. The prince discovers these things about her and is drawn to her even more. There could be a part where the prince has to choose between his royal duties and his love for the girl. But love usually wins out, and they find a way to make their relationship work within the framework of the royal world.
Often, there's a plot where the regular girl is involved in some sort of event that the prince is also attending, for example, a charity event in the town. The prince is charmed by her simplicity and kindness, and he tries to get closer to her. This can lead to a lot of challenges as the royal family might not approve of their relationship at first. But in the end, they usually overcome all the obstacles and end up together.