Entropy is a fun philosophical concept. Most of the responses so far have been from an information theory slant. I imagine most of my answer will be cosmological. Since entropy seems to be a universal constant, the idea has worked its way into the observations of researchers working in many diverse fields - evolutionary biology, psychology, fractal universe structure, etc.
The short of it is - the universe started infinitely small and will end up infinitely large. The idea of entropy is one way we think of and measure this constant, never ending process. We assume the universe is itself a closed system, so while you will see spots of low entropy here and there like planets, galaxies and message boards - the universe as a whole is moving toward a high entropy level.
It’s important to remember that entropy is one of those constant universal forces that have been with us since very soon after the Big Bang. Gravity soon got itself going also, and will always be the natural enemy of entropy. The process we think of as “time” also came about in our universe at the same… well, time.
Now fast forward to a time when physicists and chemists have evolved enough in their studies of the natural world to be asking some practical questions that they just might be able to answer with their new technology. In the mid/late 1800s a common puzzler was:
“Why can’t I build a perpetual motion machine?” We noticed it may have something to do with friction and heat dissipation.
We soon found constants and equations that would work with what we were experiencing. Much of this work was with gas pressures. We could measure and predict the “spreading out” of gas in a closed environment, and do the same things with heat energy. Look into the Boltzmann constant (gas constant/Avogadro, or even energy/temperature) for other ways in which we were also trying to find equations that would work on a macroscopic level and on the newly discovered atomic level.
So that’s why the idea of entropy came about. You will observe the natural process of entropy going on around you all the time. Real physical objects will always get colder and the clump of molecules that give it its status as physical object will all spread out until the thing is eventually dispersed elementary particles, or even smaller particles we haven’t discovered. Note that this may take a lot of time.
Thought experiment: If you take a small clear glass box and put a potted plant with a nice orange flower in it, perfectly sealed, no particles get in or out, what will happen to the object (plant) you put in there? The particles will “want” to find a state of equilibrium. The humidity level of the plant will equalize with the air, the plant’s cells will break down and turn to sludge or, if the air was dry will turn to dust. To really see the effect of entropy you would need to take this box away from as much collected energy as you can find. Out in deep space the flower’s dust particles will not be influenced even by gravity and you will see them spread out evenly in the box. Now if you open the box in deep space the particles will eventually spread out evenly across the entire universe. Everything else in the universe will also be at maximum high entropy and this will have taken, from our perspective, an infinite amount of time. After this it gets complicated.
So perhaps the most interesting philosophical aspect about entropy is that this process, as an important side effect of universal expansion, is key to our sensation of time.