Cyberpunk:

Cyberpunk is a sub-genre of science fiction, noted for its focus on “high tech and low life” and taking its name from the combination of cybernetics and punk. It features advanced science such as information technology and cybernetics, coupled with a degree of breakdown or a radical change in the social order.

Cyberpunk writers tend to use elements from the hard-boiled detective novel, film noir, and postmodernist prose to describe the often nihilistic underground side of an electronic society…. much of the action takes place online, in cyberspace, blurring any border between the actual and the virtual reality.

Classic example: Neuromancer

Steampunk:

Steampunk… concerns works set in the past, or a world resembling the past, in which modern technological paradigms occurred earlier in history, but were accomplished via the science already present in that time period.

The prototypical “steampunk” stories were essentially cyberpunk tales that were set in the past, using steam-era technology rather than the ubiquitous cybernetics of cyberpunk but maintaining those stories’ “punkish” attitudes towards authority figures and human nature. Originally, like cyberpunk, steampunk was typically dystopian, often with noir and pulp fiction themes, as it was a variant of cyberpunk.

Further derivatives: Stonepunk, bronzepunk, ironpunk, sandalpunk or classicpunk, middlepunk, clockpunk, dieselpunk and atomicpunk, transistorpunk, spacepunk… though I don’t think most of these are full-fledged subgenres, and were developed for the GURPS Steampunk role-playing game.

Classic example: The Difference Engine

Biopunk:

It describes the nihilistic, underground side of the biotech society which is said to have started to evolve in the first decade of the 21st century. Unlike cyberpunk, it builds not on information technology but on biology. Individuals are enhanced not by mechanical means, but by human genetic engineering.

Postcyberpunk: Not so much a derivative as an evolution.

Postcyberpunk describes a subgenre of science fiction which some critics suggest has evolved from classic cyberpunk. Like its predecessor, postcyberpunk focuses on technological developments in near-future societies, typically examining the social effects of an ubiquitous datasphere of computerized informaton, genetic engineering and modification of the human body, and the continued impact of perpetual technological change. Unlike “classic” cyberpunk, however, the works in this category feature characters who act to improve social conditions or at least protect the status quo from further decay.

Includes a sense of humor, as opposed to the frequently deadly serious nature of cyberpunk.

Classic example: The Diamond Age

…At some point, you know, this all seems like a snake devouring its own tail…