In so-called metaphysical poetry such as Donne's, a conceit is an elaborate metaphor in which two dissimilar things are compared over the course of a poem.
In "Death, Be Not Proud" the relevant conceit is that death is being compared to a boastful, but not very impressive, man. Donne addresses the personified figure of Death as if it were a man he'd come across in his ordinary everyday life. He subjects it to the kind of scorn normally reserved for a personal enemy, caustically mocking Death for its pride, for its pretensions to being mighty and dreadful, simply because some have called it so.
Instead, he reminds Death that it is ultimately nothing, a mere absence of life rather than something positive in its own right. For all Death's pretensions to greatness, it is barely indistinguishable from sleep. And as sleep can easily be brought on by "poppy or charms"—that is to say, drugs or medicines—there's no reason for Death to be so full of itself, so swelled with pride.