There is also such a thing as a metaphysical conceit--also a metaphor, but a comparsion between two seemingly unrelated things.
John Donne is a master at this, and an example of this is in his poem, "A Valediction: Forbidding Mourning" where he compares his love for his wife to a compass and also where he compares his love for his wife to gold.
The man is leaving his wife for a business trip and he tells her their love above all the crying and physical mourning. He says you are the fixed foot (of the compass) and I am the part with the pencil...the farther away I travel, we never really part. You just lean toward me until I come home again and we are again together.
He also compares their love to gold. He says we never leave each other--absence makes our hearts grow fonder and we are connected not just physically, but spiritually and emotionally. So, like gold, when you beat it, it never breaks. It only expands to an airy thinness like gold foil. We are like that.
Metaphors are a comparison between two things without the words "like" or "as". Usually the comparison is fairly obvious or take little thought to make the connection.
Metaphysical conceits are metaphors taken to the next level.