We will begin with the definition of "biology," the study of life. "Bio" means "life," and "-ology" is "the study of," both taken from the Greek. This is an appropriate place to start, as the formal study of life is typically traced back to the Greek academy.
Of course the study of life must go well beyond the Greeks as part of human history. It is hard to imagine that humans have not studied life since they came into existence. For example, the first archaeological record of agricultural endeavors dates back 10,000 years. It seems that agriculture would not have been possible without studying life. Yet we have no record of how that knowledge was passed on, and so biology as we know it can not be credited to that point in history. And of course most myths and creation stories are metaphors for observations of the natural world, but again it is not parallel to biology as we know it, instead it was more of a pre-cursor.
The structured approach to the study of life that we know as biology was started in ancient Greece, approximately 500 BC. This was the when the first medical school was created, and where Hippocrates and Aristotle wrote their treatises on anatomy and medicine. Other cultures at this time (like the Egyptians, Chinese, Arabs, and Persians, among others) also had sophisticated approaches to medicine, with vast knowledge of herbs and remedies, but they did not create the system to transmit this knowledge as the Greeks did, and hence lost their chance to define biology.
During the European Renaissance (approximately the 1600s) there was a surge of interest in the natural sciences, biology included with alchemy (the pre-cursor to chemistry), herbalism and medicine, and naturalism, which persists to this day as a subsidiary of biology based on observation of the natural world instead of testing.
During the 1700s and 1800s the world of biology became smaller and therefore larger. That is to say microscopes were invented and microbiology became a new field. Science was continually becoming more sophisticated and chemistry and botany became more important fields, as well as taxonomy.
In the 1900s experimental biology was beginning to emerge, defining fields such as organic chemistry, experimental physiology, cell theory, embryology, germ theory, evolution, and biogeography. These fields became stronger and even more specialized in the 21st century; the focusing in became more sophisticated in fields like molecular biology, biotechnology, and genetics, but there was also a broadening back in the spirit of the naturalists, and fields evolved such has ecology and conservation biology that focus on the bigger picture.