Metaphysical poetry is characterized by its use of conceits, extended metaphors that purport to discover similarities between two entirely dissimilar things.
In one of the most famous examples of a conceit—or infamous, depending on how one looks at it—John Donne, arguably the foremost metaphysical poet, compares a flea to the sexual relationship he wishes to have with his lover. As the flea has bitten both of them, then their blood has already been mingled together, and so there's no reason why they shouldn't go right ahead and make love.
As one can see from this example, the metaphysical conceit is designed to appeal to the intellect rather than the emotions. The conceit is a kind of philosophical argument designed to make the reader see the world around them in a completely different way. Certainly, anyone who's ever read Donne's “The Flea” will ever look at this humble parasite in the same way again.
In another of Donne's poems “The Sun Rising,” the poet uses the conceit of the sun as a kind of interfering old busybody who has the barefaced cheek to tell the speaker how to live his life. The speaker deeply resents the fact that the rising sun tells him to get out of bed as if he were a schoolboy or an apprentice.
Instead of getting out of bed to face the rigors of a day determined by the rhythms of nature, the speaker would much rather stay right where he is, in the arms of his beloved. His love for her is so much important than the rising of the sun and all that it entails. So strong is this love that it transcends the natural world and everything in it, including the mighty sun.