A nice blog post on the topic is Sean Carroll's article, "The Laws Underlying The Physics of Everyday Life Are Completely Understood" (http://blogs.discovermagazine.com/cosmicvariance/2010/09/23/...).
But even something we experience literally every moment of every day, like gravity or magnets, we haven't the foggiest idea of how it actually works. We can describe in great depth anything we wish about our models for these things and experiments can reproduce our theories with exquisite precision.
But these models are merely complex rituals, we don't actually know, at a deep fundamental level, what happens.
The Feynman answer to magnetism, for example, is deeply unsatisfying because he describes all the rituals we've discovered with science about magnetism, but never really answers the question. His answer, in 7.5 minutes are "all the electronics spin in the same direction in iron and that produces a magnified magnetic field large enough we can feel". But why do spinning electrons produce a magnetic field at all? And why does that field work the way it does?
We have built beautiful mathematical tools to describe this force, down to elementary particles, but why it exists at all is a completely mystery.
But what is your point, really?
By "understand" in physics, we really mean that we can predict what will happen in a given experiment to any desired accuracy based on the established "laws of physics".
"But even something we experience literally every moment of every day, like gravity or magnets, we haven't the foggiest idea of how it actually works."
I think this does a deep disservice to how much we actually do know about how the universe works.
Feynman also addresses this, https://www.youtube.com/watch?v=05WS0WN7zMQ
The next Monday, when the fathers were all back at work, we kids were playing in a field. One kid says to me, “See that bird? What kind of bird is that?” I said, “I haven’t the slightest idea what kind of a bird it is.” He says, “It’s a brown-throated thrush. Your father doesn’t teach you anything!” But it was the opposite. He had already taught me: “See that bird?” he says. “It’s a Spencer’s warbler.” (I knew he didn’t know the real name.) “Well, in Italian, it’s a Chutto Lapittida. In Portuguese, it’s a Bom da Peida. In Chinese, it’s a Chung-long-tah, and in Japanese, it’s a Katano Tekeda. You can know the name of that bird in all the languages of the world, but when you’re finished, you’ll know absolutely nothing whatever about the bird. You’ll only know about humans in different places, and what they call the bird. So let’s look at the bird and see what it’s doing—that’s what counts.” (I learned very early the difference between knowing the name of something and knowing something.)
Models are just a complicated way of calling something, not a way of understanding it. Getting experimental results just means you've used the correct name, not that you understand anything in particular about it.
Calling a hot bright thing "fire" doesn't mean you understand it at all -- no matter how many complex tribal dances you do around an open flame. Knowing what fire actually is gave us modern civilization. For many fundamental everyday things like gravity or magnetism, on the scale from "calling an open flame 'fire'" to "rocket ship", we're charitably somewhere around "learning to cook meat".
No! It just means that we know vast amounts about the Universe, but we still have more to learn.
So I would suggest you to begin by explicitly defining [1] what your particular notion of "understanding" is, so that you can then clarify what exactly is it that you find unsatisfying about modern physical law.
I think Feynman handles the question incorrectly. The question is about his understanding of magnetism, but he places the burden on the receiver of the explanation. If they don't understand the explanation then it doesn't matter. I do get his point in this and I'm not entirely saying it's wrong. But I think also that it's a deflection.
Lets assume a "presently omniscient" listener asking the question, somebody who may not have the knowledge before it is presented to them, but will understand the explanation perfectly as it's provided.
Suppose a Feynman or some other physicist has such a listener and they go on a deep dive into all the myriad and intricate details, the sum of all human knowledge on the matter -- this kind of listener would be able to follow along entirely with every last detail.
This listener would still not have an answer to the question "how do magnets work?" Because science doesn't actually know the answer to this. We have excellent descriptions of what magnets do, and what produces magnetism. We can predict to some nth decimal point of precision the outcome of just about any given experiment such that we don't really bother much with experimental physics about most of the day-to-day magnetism.
But why do those things produce magnetism at all?
The answers to this is basically "it just does". And that's as good as all our knowledge is. That's all we've arrived at. For all we know, every time an electron starts circling something, invisible universe fairies produce "magnetism" magic that does what we observe.
We don't know why magnetism does what it does, only that it does it.
But imagine if we actually understood it, at a fundamental level, if we "groked" magnetism and were the masters of it...
Feynman: "Magnets repel each other."
Interviewer: "What I want to know is what's going on ..."
Feynman: "Magnets repel each other."
Interviewer: "...between those two bits of metal."
Feynman: "Magnets repel each other."
Interviewer: "Well then, but what does that mean? Or why are they doing that? Or how are they doing it?"
Feynman: "Ah, [pause] uh you're asking ..."
Interviewer: "I must say that's a perfectly reasonable question to ask"
Feynman: "Of course! It's a perfectly reasonable question."
See also his quote about quantum mechanics - "There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time. There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper a lot of people understood the theory of relativity in some way or other, certainly more than twelve. On the other hand, I think I can safely say that nobody understands quantum mechanics."
https://en.wikiquote.org/wiki/Talk:Richard_Feynman#.22If_you...