OK. I'll have a go or two. Uni physics teachers would pick this to bits but don't, unless you are one.
I daresay some might go off and google to find different words as usual, without having a clue..
If there are gravitons, and the gravitons move, it doesn't mean the gravity field has moved. A field is a quantised region. Or a space in which a particle or many particles engage in the behaviour they're known for. Like gravity acts on a particle with mass.
Fields can exist with or without particles, as they mainly refer to an area of effect. However, fields related to the fundamental forces are usually used to describe the effects of particles. Those particles are recognised subatomic particles we "know" about. They're in the category of particle called bosons. Others are fermions, etc. Repeat - you can have a field with a particle, or not. The field persists when the particle is gone.
Everything is just fields. But all fields are “quantised” into highly unified bundles of energy. These bundles are called “quanta.” For example, one quantum (or bundle of energy) of the electromagnetic field is called a “photon,” and one quantum of the electron-positron field is called an “electron.” This bundling (called “quantisation”) makes it seem as though the universe is made of particles. It’s not. It’s made of quantised fields.
Fields are concepts, mathematical constructs. You can't go and get a bucketful of field. But that's all any of us is made of.
We say we reckon we understand something if we have a mathematical model for it which predicts, etc. The Higgs boson was predicted - it was required, for the Standard Model. So, yay for the Higgs boson.
The Higgs field/boson gives things mass, but, sadly but fascinatingly, not everything. It does give mass to sub-atomic particles like quarks and electrons. But it does not provide all mass to all particles, not even particles which are made exclusively of quarks. Protons and neutrons, for example, get most of their mass from the Strong Nuclear force that holds their quarks together. So, what's the rest??
Relativity explains gravity just fine. Quantum theory really doesn't in a useful way. Like, in a way, you can apply wave mechanics to a snooker ball, work out its wavelength and all that, but it's not very useful.
The effect of gravity on subatomic particles is orders of magnitude less strong than the forces holding things together down there. Not really relevant. Whether they'll extend the Standard model to encompass gravity, on the way to explaining Everything, including the far distant behaviours we ascribe to "dark" things, nobody knows.
