If we are talking "Higgs field", what is a field?
A word which signifies a space time phenomenon, that is something which can be described as a continuous function over X, Y, Z and t.
What kind of particle is it and why is it so heavy?
Have to dig into a lot of heavy specifics for that.
What does heavy mean? Massive?
Yes, around 130Gev if I recall (Giga Electron Volts, an eV being the energy an electron gains by going through a 1 volt EM field. Massive in this sense being energy (which equals mass)
Is it divisible or is that irrelevant?
Bosons break down into quarks. Leptons (like electrons) do not.
Does it occupy all of space like the ether was supposed to?
Sure, but don't fight the inverse square law.
Why would it give mass to other particles?
I don't know the specifics on this one, and it might be hard to explain anyhow.
Because it is a "field" of something molasses-likeness and pulls on or obstructs other objects with mass? How could it do that? Because it's everywhere and thus in the way? Is it really everywhere or does it have holes? Another way of saying this is it distorts space. How so? What would give it such ability?
You're trying to peek behind the curtain which isn't something physics will help you with. We say EM fields exist. Because it's a field its a continuous varying function over space time. We can describe it, manipulate it, theorize how it is before measuring and find that it is what we described, but we don't try to understand what 'it' really is. See below.
Knowing how something affects other things is one thing. HOW it does it and what it is is another. It's like saying, "You bug me. You are an annoyance." That may be true but it doesn't say what you are or how you accomplish that or what's inside of you to do that.
Yes, but physics is only interested in the 'how'. 'What' a Higgs is, in the sense you mean, isn't something that can be measured so we don't touch it. What is all about interpretation.
Well, the posts here made interesting reading. Don't know how much of a deal the Higgs makes, though.
It means that in 100 years we went from understanding about atoms to their constituent parts and behaviors. This has been a remarkable run for science, the scope of what we know about matter now is really remarkable. To be able to peek into the quantum world, as impossible seeming and as strange as it is, to this degree of sophistication is wonderful. And it means that mankind, despite budget cuts and other priorities, is still pushing the limits of our knowledge and scope.
On the face of it, it doesn't seem that grand. But you have to put it in a little context and step back to understand how remarkable this is and how far we've come.