The Dark Lantern is an important object throughout Over the Garden Wall. It is carried around by the Woodsman, who keeps it burning with oil from the Edelwood trees. It is revealed that the Woodsman believes it contains his daughter's soul, which is why he is so devoted to keeping it lit. When Greg is turning into the Edelwood tree, the Woodsman admits that he didn't know that the Edelwood trees were made of the souls of the children lost in the woods. The Beast offers to have Greg's soul put into The Dark Lantern to be kept lit, but Wirt refuses, pointing out that the Beast's obsession makes it seem like his own soul is inside it. When Wirt's suspicions are found to be true, he leaves the Lantern with the Woodsman, who blows it out and ends The Beast's existence.
This same lantern that the woodsman had was seen in the epilogue.
The Beast originally owned the lantern.
In order to keep the lantern burning, the Woodsman must bring up Edelwood branches to create oil
The Beast, many times throughout the series, claims that if the light in the lantern goes out, the Woodsman's daughter's soul will be lost forever. In actuality, she never was in the lantern and the lantern had the Beast's soul
In Chapter 10: The Unknown, the Woodsman extinguishes the lantern and the Beast is finally defeated