In its mature form, the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields. In a relativistic theory the wave function is a functional of these fields, not a function of particle coordinates. Quantum field theory hence led to a more unified view of nature than the old dualistic interpretation in terms of both fields and particles.
What is Quantum Field Theory, and What Did We Think It Is? by S. Weinberg
We have no better way of describing elementary particles than quantum field theory. A quantum field in general is an assembly of an infinite number of interacting harmonic oscillators. Excitations of such oscillators are associated with particles. The special importance of the harmonic oscillator follows from the fact that its excitation spectrum is additive, i.e. if $E_1$ and $E_2$ are energy levels above the ground state then $E_1 + E_2$ will be an energy level as well. It is precisely this property that we expect to be true for a system of elementary particles.
A. M. Polyakov, “Gauge Fields and Strings”, 1987
Undoubtedly the single most profound fact about Nature that quantum field theory uniquely explains is the existence of different, yet indistinguishable, copies of elementary particles. Two electrons anywhere in the Universe, whatever their origin or history, are observed to have exactly the same properties. We understand this as a consequence of the fact that both are excitations of the same underlying ur-stuff, the electron field.
Quantum Field Theory by Frank Wilczek
For a great “Bird's Eye View” of how Quantum Field Theory works in practice, see chapter 1 of Student Friendly Quantum Field Theory by Klauber.
For an intro to loop calculations see chapter 4 in http://physics.oregonstate.edu/~stetza/COURSES/ph654/ShortBook.pdf.
Quantum mechanics can be viewed as QFT in 0+1 dimension.
Usually in Quantum Field Theory, we compute things using a perturbative approach. There are two large problems that arise when we try to calculate things, for example the probability that two particles interact and produce a given third particle.
The probability is calculated using a perturbation series in the coupling constant, where each term in the series can be visualized with Feynman diagrams. The two problems are:
The first problem is especially problematic at a first glance. How can we trust the perturbation series even though we now that if we could calculate all loop order, the result would be infinity? The thing is that the perturbation series works perfectly up to order $O(1/\lambda)$. Then the perturbation series blows up, because it is not sensible to non-perturbative effects that must be taken into account additionally at this order.
This is described and explained perfectly with an explicit example in section 2 of https://arxiv.org/pdf/1201.2714.pdf . In addition, this problem already arises when we consider the 1-dimensional anharmonic oscillator. See, for example, this paper.
The second problem stems from the fact that we physicists do not treat distributions carefully enough. This is described in section 3 of https://arxiv.org/pdf/1201.2714.pdf.
Quantum field theoretic divergences arise in several ways. First of all, there is the lack of convergence of the perturbation series, which at best is an asymptotic series. This phenomenon, already seen in quantum mechanical examples such as the anharmonic oscillator, is a shortcoming of an approximation method and I shall not consider it further. More disturbing are the infinities that are present in every perturbative term beyond the first. These divergences occur after integrating or summing over intermediate states - a necessary calculational step in every non-trivial perturbative order. When this integration/summation is expressed in terms of an energy variable, an infinity can arise either from the infrared (low energy) domain or from the ultraviolet (high energy) domain, or both. The former, infrared infinity afflicts theories with massless fields and is a consequence of various idealizations for the physical situation: taking the region of space-time which one is studying to be infinite, and supposing that massless particles can be detected with infinitely precise energy-momentum resolution, are physically unattainable goals and lead in consequent calculations to the aforementioned infrared divergences. In quantum electrodynamics one can show that physically realizable experimental situations are described within the theory by infrared-finite quantities. Admittedly, thus far we have not understood completely the infrared structure in the non-Abelian generalization of quantum electrodynamics - this generalization is an ingredient of the standard model - but we believe that no physical instabilities lurk there either. So the consensus is that infrared divergences do not arise from any intrinsic defect of the theory, but rather from illegitimate attempts at forcing the theory to address unphysical questions. Finally, we must confront the high energy, ultraviolet infinities. These do appear to be intrinsic to quantum field theory, and no physical consideration can circumvent them: unlike the infrared divergences, ultraviolet ones cannot be excused away. But they can be 'renormalized'. This procedure allows the infinities to be sidestepped or hidden, and succeeds in unambiguously extracting numerical predictions from the standard model and from other 'physical' quantum field theories, with the exception of Einstein's gravity theory - general relativity - which thus far remains 'nonrenormalizable'.