The word "relativistic" means "compatible with principles of Special Relativity". This usually implies that we can no longer use the "classical" picture of universal stationary space and time. Instead we talk about 4-dimensional space-time.
The word "quantum" means compatible with principles of quantum mechanics. You can look them up on wikipedia etc. But essentially QM introduces a different picture of reality by separating the "information" known by the observer (in the form of wave function) from the fundamental laws of nature (in the form of operator equations).
"Field theory" and "mechanics" are different mathematical approaches to the formulation of reality. Mechanics is concerned with particles moving in space (or space-time if it is relativistic mechanics). Field theory is concerned with fields which are distributions of something over space.
Now the very non-trivial part comes: in quantum field theory you have quantum particles as quanta of your fields. This is a very remarkable mathematical fact.
You can combine all these definitions to give different theories. Of course, the most fundamental one must be relativistic, quantum and a theory of fields - QFT, the other ones being just approximations. QFT is also very likely to be an approximation of an even more fundamental theory of quantum gravity.
UPD: in practical computations the theory you should use is essentially the most simple one (or the least fundamental one) which satisfies your needs. For example, you don't need QFT to compute the motion of a baseball (classical, non-relativistic mechanics deals with it perfectly). You do, however, need Quantum Electrodynamics (which is a specific model of QFT) to compute the Lamb shift in the spectrum of hydrogen atom. No other theory explains it.
