Quantum mechanics, quantum physics or quantum theory are all umbrella terms for the branch of physics that attempts to describe the interactions of energy and matter at the atomic and subatomic scales. The term is generally used to differentiate from classical physics. Around the turn of the 20th century it was discovered that many of the laws of physics as understood at the time (classical physics) broke down when dealing with objects at high velocities, or very small objects. The theories of relativity and quantum mechanics were developed to overcome these problems. Ideally we would have a single theory encompassing relativity and quantum mechanics, however so far, nobody has been able to come up with such a theory.
These days quantum computation is often considered a subset of the broader field of study now known as quantum information science, however the term quantum information science only came into common use well after the term quantum computation (at least that is how I remember it anyway).
Generally Feynman is credited with the observation that simulating a quantum system on a classical system appears to take an exponential amount of resources, so perhaps a quantum system could perform some sort of calculation exponentially faster than is possible on a classical system. It was David Deutsch who did the pioneering work of defining a quantum computer, discussing how it would work and doing such things as defining quantum logic gates. David Deutsch did much of this work during the 1980’s. It was not until the early nineties though that quantum computation got a enormous boost when Peter Shor devised his quantum algorithm for factoring.
Since the advent of Shor’s algorithm, many people (myself included) have attempted to come up with quantum algorithms which substantially out-perform their classical counterparts, however very few have been successful.
There are really two sides to the field of quantum computation. One side takes the abstract axioms of quantum mechanics and attempts to come up with clever quantum algorithms and protocols, the other side investigates ways in which a quantum computer may be built in the future.
You might think that investigating ways of building a quantum computer would be a solely experimental pursuit, but in reality it requires contributions from both theoretical and experimental physicists. Much of my work in quantum computation was looking at simple “toy” quantum algorithms, and investigating (from a theoretical perspective) how these algorithms might be implemented using current cutting-edge experimental techniques.