It seems you can’t talk about entropy without mentioning the second law of thermodynamics. That law states that the total entropy of an isolated system can never decrease over time. This creates the asymmetry between the past and the future, the irreversibility of natural processes and the arrow of time. It is entropy that ensures that, on the macroscopic scale, time can only pass in one direction — from a state of lower entropy to one of higher entropy.
This is often simplified to define entropy as the increase in disorder with time. This is particularly favored by creationists who latch onto their own simplified version of the second law to convince themselves that evolution is impossible. Their version of the second law, usually stated something like, “Disorder increases over time,” assures them that a supernatural power is required to support life and evolution. Coupled with their mistaken belief that evolution is a force for directed improvement, this explains some of the crazy things they say.
Can you see where they went wrong in appealing to the second law? That’s right. They left out the part about where it applies to an isolated system. An open system, such as the Earth, can receive energy from an outside source, like the sun. Under those conditions the total entropy on Earth can certainly decrease, but only because the total entropy of the Earth-sun system is increasing as the sun dissipates its energy. Their other mistake is to misinterpret “can never decrease” as “always increases.” This whole process of misunderstanding and misinterpreting and misusing the second law is unironically a very good demonstration of entropy, which can never decrease in a closed mind.
The reason entropy is linked to thermodynamics is that it started out as a description of waste heat or energy loss in steam engines and other mechanical devices. Such things are never 100% efficient at turning energy into work, and the people working on the problem needed a term for their bookkeeping. It was only later as we understood more about the physics underlying thermodynamics that other definitions, such as “disorder,” evolved. It also applies to the dispersal of concentrated energy, and even the dispersal of particles.
Another way to define entropy is as the amount of energy (usually thermal energy) in a closed system that is unavailable to do work. You can have a lot of energy in a closed system — a boiler, for instance — but if the energy is evenly distributed throughout the system, then there’s no way you can get it to do work within the closed system. Therefore it has high entropy. The only way to get work out of it is to pair it up with an external system that is at a different energy level, and then tap into the energy that is transferred between them as they seek equilibrium.
Here is one more way to think of entropy. When a system is in a configuration that has few ways for its parts to be arranged, it has low entropy. A configuration that has many possible arrangements has high entropy. So a glass of water that has an ice cube in it has lower entropy than the glass of water after the ice cube has melted. In the first, all the coldest water is in the ice cube — fewer ways to do that, lower entropy. In the second, all the water is evenly distributed at the same temperature — more ways to do that, higher entropy.
So, entropy is inexorably increasing in the universe overall. It can decrease locally under the right conditions, but only at the expense of a greater increase elsewhere. It doesn’t prevent evolution, which actually depends on increasing entropy. It is entropy that tells us which way time flows — from low to high.