When we try to reduce the complexity of the world around us to understandable parts and relationships we look for patterns and then model part of the world as if that pattern explains what is happening. If I toss a rock in a pond and notice ripples on the surface, I explain the ripples in a causal manner with some kind of narrative that requires rocks thrown into ponds to lead to ripples. With such a narrative, I can try to work backwards from an observation of ripples to a statement that somehow a rock was tossed in, but there are the usual risks with working backwards with a model. There could be some degeneracy in the effects of a wide range of causes.
In the language of mathematics as it is used by physicists, we write equations for the energy and momentum of the rock and bits of fluid. We write equations for the behavior of fluids and rigid bodies tossed into them. We also use a few trial runs with the combined model to see if it works and tweak internal parameters until the predictions the model makes match well enough with reality. This kind of modeling can be done on paper or on computers by someone with a little bit of training, but at its most basic it is pretty simple. Until a model gets complicated (and they do... very fast), the mathematics doesn't get much worse than differential equations, geometry, algebra, and linear algebra. If you don't know some of these, don't freak out. You've probably used them informally and didn't know it though your experience might have been painful.
Whether one is comfortable writing physical models or not, there are some assumptions that go into them that don't involve fancy math. For example, if one models the rock tossed into the pond using momentum and energy, one has to assume the rock HAS momentum and energy, right? Energy comes in more than one form, so we might associate a few different numbers with the rock and expect them to change as it flies to its collision with the water's surface. Momentum is treated as a vector which is a number with a direction. We assign a momentum property to the rock too and expect it to change in flight. How much precision we expect for energy and momentum depends on how accurately we try to measure the flight, but we expect the rock to have well defined values for each. That is required to model the rock, right?
It turns out that there are alternate versions of these assumptions. If one models the rock as described above with well defined momentum and energy one is guilty of Classical Thinking. What this means is that we assume the rock has these properties AND that they are defined independent of anyone actually measuring them. Running the equations forward or backward in time to describe the flight path and collision speed has the built in assumption that the attributes exist. This assumption turns out to be incorrect, but physicists didn't grapple with this fact until the 20th century. On top of that there are a whole class of problems where we can safely make this erroneous assumption and still get good predictions from our models. It is as if we try to predict the positions of planets in the sky using Ptolemy's geocentric model of the universe and get reasonably good answers. The explanatory part of the model is wrong, but the rest of it works to deliver accurate results that appear not to depend on the bad assumptions.
What other assumption could be made besides the ones from classical thinking? This is where quantum thinking comes from and there are two possible versions of which I know. The first is the one we adopted and we've only recently discovered it too is wrong. The second is simply bizarre.
One break with classical assumptions is to permit the rock not to have well defined momentum and energy, but it DOES have momentum and energy. This is Early Quantum Thinking and means that a question that reads like 'how much energy does the rock have at a particular time' is not answerable absent some kind of measurement even theoretically. We must do something to the rock to discover the energy and until we do it the answer is ambiguous. That means one can't 'just run the equations.' The rock could have several values for its energy and still be what it is. It doesn't have one value until we force a situation where it must.
If you know any philosophy this should bother you because it comes very close to saying the universe is what we say it is because of what we do to it. No reasonable person trained in these quantum models would actually make that claim, but the untrained reader often makes the leap to thinking they can imagine something and make it so. There are so many who will make this leap that an unscrupulous author can make quite a bundle selling this nonsense. Those with a stronger sense of moral duty to their community will instead speak of Schrodinger's cat because that little thought experiment nicely explains the oddities that come from this way of modeling.
Unfortunately, though, it appears this is all wrong. This break with classical thinking still assumes that momentum and energy have meaning independent of observation. We model the rock as a mixed state of possible values and let an operation sort out which one actually occurs. The operations are probabilistic and would satisfy any Las Vegas bookie as impossible to fix, though one might sway them in understandable ways.
There is another way to break with classical thinking and it appears there is good experimental evidence that the universe works this way. Modern Quantum Thinking takes the next step and breaks the assumption that the attributes we measure have any meaning independent of the operation that measures them. An operation that measures the energy of the rock produces a number AND the meaning for that number. The rock itself doesn't have energy until it is measured in a way that would answer the question that asks how much energy it has. It is not that the rock has zero energy, though. It is that the energy concept is undefined. It is as if a programmer wants to read from a variable and forgot to declare it in their code earlier. The operation that answers questions about energy declares it and then measures it in one swoop. THAT is how the universe appears to work.
A whole set of reasonable questions arise now. How does one model an attribute that has no meaning until the object being modeled is measured in a way that produces the meaning AND value? How does one test such a model against reality? Can one distinguish between a successful model and one that actually explains reality? Remember the difference between geocentric and heliocentric models of the solar system. Both can be made to work, but one is very wrong! Can one write falsifiable models that work this new way? The ability to falsify models is critical to the health of science.
Modern quantum thinking can break a different assumption not mentioned above as well. We could break our assumption that phenomenon are separable. If one measurement is taken in a distant corner of the universe and another taken here, they can't influence each other as there isn't enough time for information from one to get to the other. That too appears to be wrong, but it is unclear whether this break is required or the break with meaning is required. One thing we do know is that we must either surrender separation (localization) or attribute meanings until operations occur. One of those is simply wrong and maybe both.
Is it all clear as mud now? Welcome to the world of physics. This is what some of us are pondering right now. The experimental physicists are busy in ways that occasionally hit the news sites (LHC work discovering the Higgs particle for example), but the the theorists are plugging away at the weirdest stuff you can imagine... or maybe you can't. I'm not sure I can because I'm trying to figure out how in this world I can model undefined attributes. Breaking localization doesn't bother me much, but attribute meanings too? That's asking a lot.