Conventional conductors have resistance, so when you run electrical current through them they heat up; this is why laptops and desktops have fans to cool the chips, why phones get hot and so on. Superconductors transmit energy without resistance, so the heating problem goes away and the energy costs of running the device go way, way down. Basically you can pump more through smaller wires without worrying about them melting. This means that just about any electronic/electrical device could be run more cheaply, safely, and at a smaller scale.
Advanced tech like MRI machines, maglev trains, and quantum computers all use superconductivity now, but are enormously bulky and expensive because they require extreme cooling using liquid helium (which is in short supply). Room temperature super conductors can dispense with all that, so instead of a quantum computer being the size and power draw of a refrigerator (because it is in fact mostly refrigerator) it could go in your wristwatch.
Superconductors also expel magnetic fields, which in practical terms means they repel magnets. And they only repel, without being attracted to magnets at all, like iron or the poles of other magnets. So you can use them for levitation. And because superconductors have zero resistance, if you put energy into a superconducting coil it stays there forever, just circling round and round the coil.
This LK-99 material people are talking about is an alloy of lead and copper, and it's not that difficult or expensive to make. The raw materials are fairly cheap, and the production involves heating it to hundreds of degrees centigrade for 24-48 hours, which is very easy to do in a lab and probably easy to do at industrial scale. Scientists don't understand the material very well yet, but if these discoveries are validated (as appears to be happening right now), then refining the manufacturing process is going to happen quite quickly because the payoffs and economic demand will be enormous.
People are comparing this to the invention of the transistor; I think a better comparison is the electrical lightbulb. It's going to change things massively, because any country will be able to manufacture this. You could manufacture this stuff at home, the equipment you need fits on a desk and costs only a few thousand $.
There's no general analytical method to solve for a material with specific properties, including superconductivity. The properties of materials are governed by the quantum mechanical behavior of electrons and their interactions with the crystal lattice. These interactions are typically described by the laws of solid-state physics, which involve solving complex quantum mechanical equations for the electrons in a crystalline structure.
For simple systems, such as some low-temperature superconductors and idealized models, researchers can sometimes make analytical approximations or derive simplified equations that describe the behavior of the material. But for most materials, the equations are too complex to solve analytically, and researchers rely on numerical methods, computational simulations, and empirical data to understand and predict material properties.
It's similar to the reason we can't find easy analytical solutions to other complex systems (like models trained via Machine Learning), there are just too many complex factors and interactions to take into account.
Same reason a photograph of a delicious meal (the theorized characteristics of the material you want) doesn't give much idea of how to make it, even if you have a stove and live next to a supermarket with all the ingredients.
The cost of the cooling subsystem and whatever workarounds you need to do to deal with heat buildup, plus about 10-20% in resistivity iirc. Not a pro engineer, but have some experience with building electronics and embedded systems.
Everything that involves powerful magnets and benefits from making things float - like bearings.
Robots and exoskeletons come to mind.
One of the more bonkers applications would be to wrap Mars around the equator with it, creating an artificial magnetosphere.
Temperatures on Mars are pretty low, but during the Martian summer they get to a nice 20°C there, so currently available superconductors are not up to the task.
MRI machines will get much better/cheaper. They use superconductors already, but are very hard and expensive to keep at their super-cold operating temperature.
Can someone ELI5 why this matters? What does it enable technologically?
Flying cars? Fusion reactors? Magic fix for global warming?