Partly because of LessWrong discussions about what really counts as understanding (some typical examples), I came up with a scheme to classify different levels of understanding so that posters can be more precise about what they mean when they claim to understand -- or fail to understand -- a particular phenomenon or domain.
Each level has a description so that you know if you meet it, and tells you what to watch out for when you're at or close to that level. I have taken the liberty of naming them after the LW articles that describe what such a level is like.
Level 0: The "Guessing the Teacher's Password" Stage
Description: This level is only included for comparison -- to show something that is not understanding. At this point, you have, a best, labels that other people use when describing the phenomenon. Maybe you can even generate the appearance of understanding on the topic. However, you actually have a maximum entropy probability distribution. In other words, nothing would surprise you, no event is more or less likely to happen, and everything is consistent with what you "know" about it. No rationalist should count this as an understanding, though it may involve knowledge of the labels that a domain uses.
Things to watch out for: Scientific-sounding terms in your vocabulary that don't correspond to an actual predictive model; your inability to say what you expect to see, and what you would be surprised by.
Level 1: The "Shut up and Calculate" Stage
Summary: You can successfully predict the phenomenon, but see it as an independent, compartmentalized domain.
Description: This is where you can predict the phenomenon, using a generative model that tells you what to expect. You are capable of being surprised, as certain observations are assigned low probability. It may even be tremendously complicated, but it works.
Though low on the hierarchy, it's actually a big accomplishment in itself. However, when you are at this stage, you see its dynamics as being unrelated to anything else, belonging to its own domain, following its own rules. While it might have parallels to things you do understand, you see no reason why the parallel must hold, and therefore can't reason about how extensive that relationship is.
Things to watch out for: Going from "It just works, I don't know what it means" to "it doesn't mean anything!" Also, becoming proud of your ignorance of its relationship to the rest of the world.
Summary: Your accurate model in this domain has deep connections to the rest of your models (whether inferential or causal); inferences can flow between the two.
Description: At this stage, your model of the phenomenon is also deeply connected to your model of everything else. Instead of the phenomenon being something with its own set of rules, you see how its dynamics interface with the dynamics of everything else in your understanding. You can derive parameters in this domain from your knowledge in another domain; you can explain how they are related.
Note the regression here: you meet this stage when your model for the new phenomenon connects to your model for "everything else". So what about the first "everything else" you understood (which could be called your "primitively understood" part of reality)? This would be the instinctive model of the world that you are born with: the "folk physics", "folk psychology", etc. Its existence is revealed in such experiments as when babies are confused by rolling balls that suddenly violate the laws of physics.
This "Level 2" understanding therefore ultimately connects everything back to your direct, raw experiences ("qualia") of the world, but, importantly, is not subordinate to them – optical illusions shouldn't override the stronger evidence that proves to you it's an illusion.
Things to watch out for: Assuming that similar behavior in different domains ("surface analogies") is enough to explain their relationship. Also, using one intersection between multiple domains as a reason to immediately collapse them together.
Level 3: The "Truly Part of You" Stage
Summary: Your models are such that you would re-discover them, for the right reasons, even they were deleted from your memory.
Description: At this stage, not only do you have good, well-connected models of reality, but they are so well-grounded, that they "regenerate" when "damaged". That is, you weren't merely fed these wonderful models outright by some other Really Smart Being (though initially you might have been), but rather, you also consistently use a reliable method for gaining knowledge, and this method would eventually stumble upon the same model you have now, no matter how much knowledge is stripped away from it.
This capability arises because your high understanding makes much of your knowledge redundant: knowing something in one domain has implications in quite distant domains, leading you to recognize what was lost – and your reliable methods of inference tell you what, if anything, you need to do to recover it.
This stage should be the goal of all rationalists.
Things to watch out for: Hindsight bias: you may think you would have made the same inferences at a previous epistemic state, but that might just be due to already knowing the answers. Also, if you're really at this stage, you should have what amounts to a "fountain of knowledge" – are you learning all you can from it?
In conclusion: In trying to enhance your own, or someone else's, understanding of a topic, I recommend identifying which level you both are at to see if you have something to learn from each other, or are simply using different standards.