Contributed by Daniel Daniel
To put it simply, biomimetics is the study of design principles in biological systems with the view of integrating them in engineering systems and modern technology. In some sense, biomimetics can be viewed as a process of reverse-engineering of biological systems. This is often a fruitful exercise, because evolutionary pressures often forces living organisms to be highly optimized and efficient. There are many early examples of biomimetics, such as the invention of velcro, which was inspired by tiny hooks found on the surface of burs and the cat's eye reflectors which were the results of studying the mechanism of cat's eyes.
Examples of biomimetic systems can be found in the wikipedia article on bionics. http://en.wikipedia.org/wiki/Bionics
From the point of view of chemistry, biological systems are able to synthesize complex chemical compounds efficiently at relatively low temperature (e.g. human body's temperature ~37 degrees celsius), whereas we often requires the use of high temperature, high energy and huge reactors. Biological systems often achieve these through enzymatic reactions and it will interesting to study the way biological systems snynthesize chemical compounds to better optimize the way we do chemistry.
A good article discussing this by Ronald Breslow in the Journal of Biological Chemistry. http://www.jbc.org/content/284/3/1337.full
Difference between biological systems and artificial systems
One main difference between biological and artificial system is that the former is responsive to the environment, exhibit homeostasis and self-repair properties, while the latter is often static, lacks self-regulatory abilities and is relatively unresponsive. The interest in studying biological systems is in part hoping to incorporate their design principles in smart material in the future which can responds to different environments appropriately. Examples include glass windows that can regulate the amount of sunlight entering the room to optimize energy efficiency.