What is the Higgs?

Very good presentation for someone like me with no knowledge in physics but still facinated by all those discoveries! :) Thanks

Nicely made, however most visible mass (i.e. the mass of protons/neutrons) is generated dynamically by the quark interactions.

No, the Higgs boson has nothing to do with gravitational force - only mass. There's not yet an accepted theory on how the gravitational force exist, but researchers are looking for a spin-2-boson called a Graviton to explain it. More reading: https://en.wikipedia.org/wiki/Graviton

I have a basic understanding of electrons and the electric field. Can I think about a Higgs particle like a particle with a mass charge? Does the field work similarly to the electric field? Do all particles with mass have a Higgs particle in them?

To answer your questions: 1) sorta. "mass charge" is just mass. In quantum field theory, particles can have an intrinsic mass parameter. The higgs mass parameter is interesting in that it is imaginary (i.e. its value squared in negative).

2) Not really. In quantum field theory, the fundamental spacetime entity is a field, and quantized excitations of the field are interpreted as particles. So the "Higgs field" is just a postulated field in spacetime, which is associated with the observable particle called the Higgs Boson.

3) No. Roughly speaking, the mathematical mechanism by which the Higgs "gives particles mass" is like this: You write down the theory with the Higgs particle doing its own thing; it turns out this theory can be "viewed" from different perspectives (physicists refer to this as a gauge). If you view the theory from a certain perspective (see: spontaneous symmetry breaking), the algebra of the equation gets all jumbled up. When you untangle the equation by factoring etc, you get back to something that looks like a physics theory, but now the Higgs got mixed into various places in the equation. In some places, it looks just like a "mass charge". This mechanism was originally proposed to get around a nasty problem in physics; our super-awesome theory of electroweak interactions was very clear that the weak bosons (W and Z) must be massless. But when we discovered these particles they were really heavy. The Higgs mechanism is a loophole that allows these bosons to be formally massless in the theory, but which get mixed with the Higgs to give an observable mass as described.

The whole idea of a field having "imaginary mass", and that our universe could be in a spontaneously broken vacuum state (i.e. that we're viewing the theory from a weird perspective) are pretty hard to swallow. This is while the discovery of the Higgs boson is a huge vindication for theoretical physics; it puts all these crazy-sounding ideas on empirical footing.