There are many different ways of conceptually approaching
quantum field theory. I find that the
most satisfying way of understanding quantum field theory is to
think of it as a cross between special relativity and quantum mechanics.

Quantum Mechanics deals with the behavior of some given
number of particles which may interact with each other and/or with some external
forces. The quantum theory that you’re
interested in (This includes the number of particles, all external forces, and
the way they interact) can usually be represented by a mathematical operator
known as the Hamiltonian. For example,
your Hamiltonian might describe the interaction between an electron and a proton,
or a single electron sitting in a magnetic field.

What special relativity adds to this is the notion that
matter and energy are one and the same, (most readily demonstrated by Einstein’s
famous formula E=mc^{2}) and therefore particles can be created or
destroyed (at a cost of some amount of energy), so that the number of particles is no longer a constant in your
theory. Therefore, if we want an
accurate picture of quantum mechanics which obeys the laws of special
relativity, it is necessary to change how we construct our theory. For example, when you turn on a light switch,
you’re creating zillions of photons.
There is no theory in nonrelativistic quantum
mechanics which can describe this.

Conceptually, what Quantum Field
Theory does to solve this problem is it “combines” all Quantum theories with N
particles, to get a general theory with all possible combinations of particles. Mathematically, it expands your Hilbert Space
(the space of all possible quantum states with N particles) into a much bigger
space, called Fock Space, the space of all possible
quantum states with any combination of the particles in your theory.

The cumbersome mathematics of
Quantum Field Theory (e.g. infinite-dimensional path integrals) can be
radically simplified by making use of what are known as Feynman diagrams. A simple way of understanding a Feynman
diagram is that it represents a possible path a system could take to get from
one configuration to another. For
example, for a photon to get from point a to point b, it might just go
directly from a to b without doing anything, or it might split into an electron
and a positron, then the electron and positron could collide, annihilating
one another, and producing another photon, which propagates the rest of the
way to point b. Or this electron-positron
process could happen twice. These are
three of the infinite possible diagrams which contribute to the process of the
photon propagating from point a to point b.

a----------------b

a----<>---------b

a----<>---<>---b

For Feynman Diagrams that don’t
look like total crap, See Peskin & Schroeder’s *An Introduction to Quantum Field Theory*,
or pretty much any textbook on the subject.

One interesting prediction
Quantum Field Theory makes is the presence of virtual particles, which appear
and disappear out of nowhere; they require no energy cost to appear (i.e. they can violate conservation of energy), as long
as they disappear sufficiently quickly.
Feynman diagrams for these particles have no endpoints like the diagrams
above; they appear only as a combination of connected loops.

While quantum field theory has
some problems (regularization and renormalization procedures generally have you adding and
subtracting infinities in ways that would make most mathematicians lose
their cookies), it
has provided the most accurate and precise description of particle physics to
date. If the quantum theory you’re
studying is the standard model, it is possible to accurately predict the electromagnetic
force, strong nuclear force, and weak nuclear force. Generalizing to quantum field theory in curved
spacetime, it is possible to predict the interaction
of said forces in an external gravitational field, however a way has not yet
been discovered to “quantize” this gravitational field properly, and so it must be considered as an external force, thus gravitons must
still remain a mystery to particle physicists, at least until string theorists can shed some light on the subject.