A set of foolish questions

Found an answer for 3 / 4... In physics, a field equation is a mathematical statement describing how the fundamental forces interact with matter and energy. The four fundamental forces are gravitation, electromagnetism, the strong interaction and the weak interaction.

Boy, "in simple terms"? Quite the challenge. But I will try.

1. Okay, we're familiar with the idea of "relativity", right? The laws of physics should stay the same from any uniformly moving point of view. (The difference between Newtonian and Einsteinian is the speed of light limitation, but the idea is the same). Technically, it means that the laws of physics are "invariant upon symmetry transform", i.e., a simple motion translation leaves the laws unchanged. In Yang-Mills theory, laws of physics are likewise [should be] invariant upon certain local symmetry transforms, called gauge transformations, which you can try to imagine to be local spacetime deformations rather than global. The consequences of this invariance create the "Yang-Mills Field", i.e., it behaves as-if there are gauge bosons, which extends quantum electrodynamic field theory to include both the strong and weak force. This is the foundation of the particle Standard Model, which would not be possible with QED alone.

2. Loops in Feynman diagrams are quite literally particles going around in a loop, usually through time. As probably the simplest example, a positron can be treated as an electron going backwards in time, so when we have a virtual pair production, i.e., given a burst of quantum energy, an electron-positron pair can be produced, which is usually short-lived, recombining a short time later (VERY short!), producing energy which was "borrowed" just a moment ago to create the pair. But this can alternatively be seen as just one electron going around in an endless loop through time. I know it sounds so fantastical and weird, but it all works out mathematically. Needless to say, loops like this create nothing but grief and trouble for the mathematicians who have to try to work out the numbers, but, nevertheless, real numbers come out of them, and are verified by experiment.

3. Ah, the best way to describe a "field" is to compare it with "particles". A charged particle, for example, "creates an electric field", but it can equally be said that it is the field itself that "creates the particle". Get it? You have a choice of looking at it either way. Quantum field theory gives you the tools to make that choice. Another way of looking at it, if the rest of the universe did not exist, could we have an electron? Many physicists say no. As an interesting analogy, there's Mach's Principle, which says that if the rest of the universe did not exist, then we couldn't have centrifugal force.

4. A field equation differs from a classical equation is that with the latter, we work with "particles", but with the former, it's more "waves", i.e. everything is extended through all spacetime. Quantum field theory says one cannot understand and analyze particles in isolation from everything else. This is particularly true if one attempts to understand the attributes of particles, as described in the Standard Model, i.e., masses, charges, spin, interactions, etc. Imagine that a deaf and blind man was given hearing and sight---suddenly the world seems like a really noisy and chaotic place. That's the rough analogy between classical mechanics and quantum field theory. As a very famous example, just recently the existence of Higgs Boson was verified at CERN. Higgs Boson is what "gives things mass", but another way of looking at it is that there is a "Higgs Field", the consequence of which is that things, like you and I, have mass. In other words, once again, if the rest of the universe did not exist, there would be no Higgs Field, and we would be massless, and likely not exist at all and having this conversation.