Neuroscience encompasses molecular
systems and Cognitive
It is defined as any of the sciences that deal with the nervous system, such as neuroanatomy or neurobiology.
One could argue that the science has existed ever since it was observed that people don't operate so well without their brain. However, much of the methodical observations that have evolved into neuroscience have been within the last 400 years or so.
In 1649, Rene Descartes considered the pineal gland as the center of the mind - the connection between your spirit and your meat. Antony von Leeuwenhoek observed nerve fibers in flesh in 1717, and 74 years later Luigi Galvani was stimulating frogs nerves with electricity.
Things really start moving for biology in 1839 with Theodor Schwann's cell theory - observing that cells are, in fact, the fundamental atoms of animal and plant life. It won't be until 1889 that Otto Friedrich and Carl Dieters are able to get into some real neural cellular biology and differentiate the dendrite and axon of a neuron. By 1855, Bartolomeo Panizza had demonstrated that the occipital lobe was essential for vision.
Closer to the turn of the century, Camillo Golgi sought a connection between the body and mind - between neuroscience and psychiatry. Electrical activity is recorded in the brain around 1875 by Richard Caton, and in 1891, Wilhelm von Waldeyer coins the word neuron. Five years later, Rudolph Albert von Kolliker coins the word axon.
In the early 1900s, Harvey Williams Cushing sucessfully uses electricty to stimulate the human sensory cortex. Work on the function and structure of nerve cells wins the Nobel Prize for Santiago Ramón y Cajal and Camillo Golgi, jointly. By 1919 Cecile Vogt can describe over 200 cortical areas in the brain. In 1924, Hans Berger creates the first EEG and discovers and explores brain waves.
Michael Stipe gets a boon in 1953 when Nathaniel Kleitman and Eugene Aserinsky describe rapid eye movement. In 1949, Donald O. Hebb published his book The Organization of Behaviour, wherein he would put forth a theory of behaviour with a physiological basis, giving the connectionists something to do later on. Sir John Carew Eccles, Sir Alan Lloyd Hodgkin and Sir Andrew Fielding Huxley win a Nobel Prize in 1963 for discovering the chemical mechanism by which the impulses between nerve cells are moderated.
This field is now wide and widespread; it even has its own society.