In contrast to many other physical theories there is no canonical definition of what QFT is. Instead one can formulate a number of totally different explications, all of which have their merits and limits. One reason for this diversity is the fact that QFT has grown successively in a very complex way. Another reason is that the interpretation of QFT is particularly obscure, so that even the spectrum of options is not clear. Possibly the best and most comprehensive understanding of QFT is gained by dwelling on its relation to other physical theories, foremost with respect to QM, but also with respect to classical electrodynamics, Special Relativity Theory (SRT) and Solid State Physics or more generally Statistical Physics. The connection between QFT and these theories is also complex and can not be neatly described step by step.
If one thinks of QM as the modern theory of one particle (or, perhaps, a very few particles), one can then think of QFT as an extension of QM for analysis of systems with many particles-- and therefore with a large number of degrees of freedom. In this respect going from QM to QFT is not inevitable but rather beneficial for pragmatic reasons. A general threshold is crossed when it comes to fields, like the electromagnetic field, which are not merely difficult but impossible to deal with in the frame of QM. Thus the transition from QM to QFT allows treatment of both particles and fields within a uniform theoretical framework. (As an aside, focusing on the number of particles, or degrees of freedom respectively, explains why the famous renormalization group methods can be applied in QFT as well as in Statistical Physics. The reason is simply that both disciplines study systems with a large or an infinite number of degrees of freedom, either because one deals with fields, as does QFT, or because one studies the thermodynamic limit, a very useful artifice in Statistical Physics.) Issues regarding the number of particles under consideration yield yet another reason why we need to extend QM. Neither QM nor its immediate relativistic extension with the Klein-Gordon and Dirac equations can describe systems with a variable number of particles. Obviously this is essential for a theory that is supposed to describe scattering processes, where particles of one kind are destroyed while others are created.
One gets a very different kind of access to what QFT is when focusing on its relation to QM and SRT. One can say that QFT results from the successful reconciliation of QM and SRT. In order to understand the initial problem one has to realize that QM is not only in a potential conflict with SRT, more exactly: the locality postulate of SRT, because of the famous EPR correlations of entangled quantum systems. There is also a manifest contradiction between QM and SRT on the level of the dynamics. The SchrÃ�Â¶dinger equation, i.e. the fundamental law for the temporal evolution of the quantum mechanical state function, can not possibly obey the relativistic requirement that all physical laws of nature be invariant under Lorentz transformations. The Klein-Gordon and Dirac equations, resulting from the search for relativistic analogs of the SchrÃ�Â¶dinger equation in the 1920s, do respect the requirement of Lorentz invariance. Ultimately they are not satisfactory because they do not permit a description of fields in a principled quantum-mechanical way.
For various phenomena it is legitimate to neglect the postulates of SRT, namely when the relevant velocities are small in relation to the speed of light and when the kinetic energies of the particles are small compared to their mass energies mc2. And this is the reason why non-relativistic QM, although it can not be the correct theory in the end, has its empirical successes. It can never be the appropriate framework for electromagnetic phenomena because electrodynamics, which prominently encompasses a description of the behavior of light, is already relativistically invariant and therefore incompatible with QM. Scattering experiments are another context in which QM fails. Since the involved particles are often accelerated almost up to the speed of light, relativistic effects can no longer be neglected. For that reason scattering experiments can only be correctly grasped by QFT.
The catchy characterization of QFT as the successful merging of QM and SRT has its limits. On the one hand, as already mentioned above, there also is a relativistic QM, with the Klein-Gordon- and the Dirac-equation among their most famous results. On the other hand, and this may come as a surprise, it is possible to formulate a non-relativistic version of QFT (see Bain 2011). The nature of QFT thus can not simply be that it reconciles QM with the requirement of relativistic invariance. For a discriminating criterion it is more appropriate to say that only QFT, and not QM, allows describing systems with an infinite number of degrees of freedom, i.e. fields (and systems in the thermodynamic limit). According to this line of reasoning, QM would be the modern (as opposed to classical) theory of particles and QFT the modern theory of particles and fields. However, and this shall be the last turn, even this gloss is not untarnished. There is a widely discussed no-go theorem by Malament (1996) with the following proposed interpretation: Even the quantum mechanics of one single particle can only be consonant with the locality principle of special relativity theory in the framework of a field theory, such as QFT. Ultimately, the characterization of QFT, on the one hand, as the quantum physical description of systems with an infinite number of degrees of freedom, and on the other hand, as the only way of reconciling QM with special relativity theory, are intimately connected with one another.