Biopunk is a general name for a sub-genre (or two) of science fiction in which new and interesting creatures can be created by human scientists. Its meaning is highly fluid, and must be determined based on context.
Biopunk may be used in the same sense as are dieselpunk, steampunk, and clockpunk. In this case, it would refer to a society or alternate history in which modern technologies have not been developed, but the biological sciences have moved beyond what we have today -- generally through pseudo-magical efficiency in the fields of medicine, genetic engineering, or hybridization. H. G. Wells's The Island of Dr. Moreau and Mary Shelley's Frankenstein would be the canonical examples. Perhaps because of these examples, Biopunk may occasionally be used to refer to steampunk stories that include fantasy monsters (vampires and werewolves particularly).
However, it is more common to use biopunk to mean something akin to cyberpunk or even transhumanism SF; stories about our future where humans can be easily modified into any form, where computers are grown in vats to perform functions through protein folding, or where spaceships are genetically engineered living creatures. This may sometimes also be referred to as Ribofunk or biohacking. The term biopunk, in this case, would have the emphasis on the 'punk', generally indicating a darker, gritty type of story with strong social themes and an strong focus on one type of technological advancement -- obviously, genetic manipulation in this case.