A metaphor compares one thing to another but it does not use the words, and it does not use like or as to compare, as a simile would. A metaphor is a little like written magic.  The word metaphor comes from metaphor-mo-sis, which means to transform.

Here is the Merriam-Webster dictionary definition of metaphor:
met·a·phor
noun
a word or phrase for one thing that is used to refer to another thing in order to show or suggest that they are similar
an object, activity, or idea that is used as a symbol of something else Merriam-Webster
“all the world’s a stage”

“Tomas did not realize at the time that metaphors are dangerous. Metaphors are not to be trifled with. A single metaphor can give birth to love.”
― Milan Kundera The Unbearable Lightness of Being

A metaphor compares one thing to another but it does not use the words, and it does not use like or as to compare, as a simile would. A metaphor is a little like magic in writing.

Popular metaphors

800-pound gorilla
The elephant in the room
A heart of stone
Apple of my eye
Cabin fever
Fog of war
The exam was a breeze
John is an old flame of hers

While metaphors are part of everyday language, extended metaphors are very common in poetry, song lyrics, and literature. An extended metaphor is a technique by which an already existing metaphor is carried over into the next sentence, verse, paragraph or even throughout a particular piece writing.

In Katy Perry’s pop song “Firework,” the singer uses an extended metaphor to compare an insecure person who is waiting to shine to a firework waiting to brighten the night sky:
“Cause baby, you’re a firework
Come on show them what you’re worth
Make them go, “Oh, oh, oh”
As you shoot across the sky.”

If you want to make any subject more interesting for a reader, metaphors are a wonderful way of transporting someone’s imagination.  And of course, for all those poets, songwriters and dreamers out there, metaphors are the air they breathe.
The magic of metaphor is described in this quote by José Ortega y Gasset  The Dehumanization of Art and Ideas About the Novel
“The metaphor is perhaps one of man’s most fruitful potentialities. Its efficacy verges on magic, and it seems a tool for creation which God forgot inside one of His creatures when he made him.”