There is a way of understanding irreversible processes that has always amused me: suppose we put sand of two colors in a container. First we put the color 1 on the bottom. Then, on top, the color 2 is poured. Clearly, the sand is ordered, each color in its place. Now, we take a rod or a stick and we stir in a clockwise direction. After a certain time, all the sand will have been mixed and it will be impossible to distinguish the two colors: now only one more or less uniform color remains. If this mixing process were reversible, it would be enough to stir the sand with the stick again, this time turning it counterclockwise, and we could see how the colors separate again. It's not like that, is it? Well, this is an example of an irreversible process.
Another example, where we can see that processes in which entropy decreases, do so at the expense of energy consumption: Let's consider my office... Suppose (which is a lot to assume, but for the purposes of the example we'll pretend that were possible) that today everything is perfectly ordered. That is to say, each book in its place, magazines and papers properly placed in their trays or filing cabinets, collected computer cables, sharpened pencils and closed ballpoint pens placed in their cups... etc. OKAY? Well, as is easy to suppose, if we let time pass, the entropy will only increase constantly. That is to say, that the spontaneous process of habitual activity means that after a few days (even a few hours) all that is in a certain degree of disorder (to put it finely...). The process should not be confused with a reversible process. Of course, we can put everything back in its place, but it will be at the expense of expending large amounts of energy needed to put everything back in its place.
We can therefore distinguish two classifications for processes: one would be reversible/irreversible and the other would be spontaneous/non-spontaneous.
A process is reversible when ΔS = 0. As you can easily think, Murphy's law already warns us that this type of process is virtual... Where has it been seen that things start spontaneously and without energy consumption? fix alone? what not? Well, that... Therefore, the vast majority of observable processes in the Universe are all irreversible, that is, the entropy does not stop increasing.
On the other hand, the classification into spontaneous and non-spontaneous processes gives more play... A process is spontaneous when it does not need energy to start, that is, when its activation energy is zero. Indeed... almost without spending any energy my office messes itself up (what things, right?). On the other hand, the process is non-spontaneous when a little push is needed to start it. That energy that we have to give for the process to start is the activation energy. This should not be confused with the fact that the process is exoenergetic or endoenergetic, a common mistake in novice students. My example is a clear case of an irreversible, non-spontaneous and quite endoenergetic process. Ha ha ha!