Let me try. A Quantum Field Theory(QFT) attempts to find the PROBABILITIES of various kinds of interactions between particles. The basic objects it deals with however are fields, more or less complicated ones, extending throughout space, and the particles are viewed as excitations ("quanta") of these fields.
Richard Feynman made two contributions to QFT: Feynman diagrams and path integrals. Feynman diagrams are drawings of the interactions between the particles. A very basic diagram looks like an H; two particles come up from the bottom, exchange a force particle (the crossbar of the H) and are thereby transformed into some other particles, perhaps, and the two outgoing particler exit out the top. So you break down this drawing and it has five lines (bottom two legs, crossbar, top two legs), and two vertices (the ends of the crossbar, where it meets the other legs). And Feynman showed how to associate each of these seven items with a mathematical expression and put the expressions together to form an integral, which integral gives the basic probability for the interaction that the diagram describes. Think that over a bit, it's really not hard. Of course most particle interactions are much more complicated and the mathematics can get really long and intricate.
A word about the perturbation approach. This means starting with the simplest diagrams, calculating the probability, then going to the next simplest, calculating and adding these two results together, and so on. You get an infinite series which hopefully, although it may not converge, gives good answers for the first few terms. You'll see the terms referred to as "n-loop" meaning a diagram in which n photons (or other particles) are emitted and then reabsorbed; this produces a loop in the diagram; and these loops are the principle complication of the diagrams, and of the probabiity calculations. The theory called Quantum Electrodynamics has been around for nearly sixty years and they are just now getting into five loop calculations.
This then gives a classical, unquantized theory, and Feynmann's other contribution, path integrals, is one of the ways to make it into a quantized theory. I am not going to describe this, becaise Feynman has done it far better in a little book for intelligent but not mathematically sophisticated people that he calls QED. I most strongly suggest that you read it.
But I will say this, all QFTs have a problem. At very short distances, really close to the particles themselves, those probability integrals go to infinity. Physicsts have learned that this is because any QFT they have is only an "effective theory"; it correctly describes physics at some energy scale, but cannot be expected to give meaningful results at a much higher energy scale. In quantum land, small distance means high energy, so that, the physicsts tell themselves, is the reason for the infinities.
This gives them a strategy for dealing with the infinities; they apply a smooth function which turns off the integral at very short distances; this is called a Regulator and after a little work it shows up as an undetermined constant in the integral. Depending on the theory you may have several of these constants. After you quantize and use the integrals to calculate probabilities, you will have various quantities like the masses of the particles that the theory doesn't predict and you can combine the undetermined constants into these "counterterms" formally and get rid of them. Then you just plug the measured values of those quantities into the integrals. This is called Renormalization. Then you have a set of integrals that give finite probabilities, and the ones we know best are very very good at that.
Now we come to gravity. When you do all this with general relativity, you get an infinite number of undetermined constants, and you don't have an infinite number of counterterms to absorb them. So we say that general relativity cannot be renormalized, and therefore a valid QFT of it cannot be derived.
