What is the link between metaphysical poetry and a conceit? What do these terms mean?

A conceit is a comparison--a simile or metaphor, for instance--of a complicated nature in which the the poet deliberately chooses two unlike things and then proceeds, often in a very elaborate way, to demonstrate a likeness between them. This is what the "metaphysical" poets such as Donne, Cowley, George Herbert, Andrew Marvell and others specialized in. In "A Valediction: of Weeping," Donne begins:

"Let me pour forth
My tears before thy face, whilst I stay here,
For thy face coins them, and thy stamp they bear."

Few poets other than Donne would have thought of likening tears (in which the reflection of his girlfriend's face is seen) to coins. Similarly, in the following,

"Mark but this flea, and mark in this,
How little that which thou deniest me is;
It sucked me first and now sucks thee,
And in this flea our two bloods mingled be."

The idea of a flea representing the union of a man and woman would hardly be considered passionate in a conventional sense. (Note also in these lines that in 16th- and 17th-century printing the initial "s" and the "f" looked similar.) Not only is it an unlikely comparison, which Donne then elaborates more fully, but it's typical of Donne's rather impudent, iconoclastic view of love that is expressed throughout his work.
John Dryden is usually credited as the first to apply the term "metaphyisical" to Donne and the others, as a criticism. Samuel Johnson later used the term with a more detailed description of these poets and why their manner of expression was unacceptable, was not concordant with the ideals of English poetry from the time of the Restoration (1660) on. The term implies something "beyond" what is considered normal or natural for poetry to express, as their conceits in particular were alleged to do. Interestingly, however, Johnson in his "Life of Cowley" gives so many quotations from these poets that he must have read volumes of their work, and therefore appreciated their genius, as we do now. Therefore today the term "Metaphysical Poets" merely identifies them, without any value judgement--which for most readers from the early twentieth century on has been a positive one in any event.