The Holly King and the Oak King are part of Celtic mythology, and they represent two sides to the Greenman, or Horned God.
They battle twice a year, once at Yule and once at Midsummer (Litha) to see who would rule over the next half of the year. At Yule, the Oak King wins and at Litha, the Holly King is victorious. In other words, the Oak King rules over the lighter half of the year, and the Holly King over the darker half. The change from one to the other is a common theme for rituals at Yule, and also at Midsummer.
Another version of the Holly King and Oak King symbolism, is that they do not directly switch places twice a year, but rather both live simultaneously. The Oak King is born at Yule, and his strength grows through the spring, peaks at Beltane and then he weakens and dies at Samhain. The Holly King lives a reverse existence, and is born at Midsummer, waxes more powerful through the summer and fall, to his peak at Samhain.
His influence then lessens until Beltane, when it is his turn to pass away. In this perspective, the two Kings enjoy a more intricate interplay of power and is perhaps a better illustration of their duality. At any given time, they both exist but have varying levels of influence throughout the year.
Either way, each King represents different ideas. The time of the Oak King is for growth, development, healing, and new projects. The Holly King’s time is for rest, reflection, and learning.