A metaphor is a comparison between two things that share an important trait. Unlike similes, metaphors do not use the words "like" or "as" to make the comparison. Instead, they imply a similarity by saying one thing is another, like this: "The sky is a fried egg."
In this metaphor, the sky is being compared to a fried egg because both of them, perhaps, are white with an immense, round, yellow center. The sky might also be hot. This is the simplest kind of metaphor to see because the equation ("something is something else") is spelled out by the author. However, sometimes a metaphor is harder to see, as in this sentence: "My heart burns for you." Here, heart is compared with fire, but the author does not explicitly say, "My heart is on fire." Instead, she says that her heart burns, an action that applies to fires, not hearts.