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Let's say you have a balloon. You can blow up this balloon to it's full size, and then you can let all the air out of it until its back to its deflated self. You can blow the balloon up again if you want - but what if it pops? Can you get the balloon back to what it was before? Highly unlikely! We can look at blowing up and deflating the balloon as reversible processes - but popping it is an irreversible process! Now let's apply this to thermodynamics. It gets a litle more complicated!
The formal definition of a reversible process is a process in which the system and surroundings can be restored to the initial state without producing any changes in the thermodynamics properties of the universe. Phew! What does that mean? Basically it says that the state of the system is always in equilibrium, even when it's undergoing a change. If this sounds fishy to you, then you would be correct! Reversible thermodynamic processes do not happen in nature; they are merely tools to make calculations easier for phycisists. In reality, processes would have to happen infinitely slowly for them to be truly in equilibrium at all times! Reversible process are ideal; there are no extraneous factors like heat and fluctuations in entropy to deal with. For example, the work done by a system during a reversible process is the maximum work that can be done.
All real processes are irreversible. This means that in going from the initial state to the final state, some change in the thermodynamic properties of the universe occurs. Which one is it? Well, heat is a form of energy, and we know that the law of conservation of energy holds steady. However, according to the 2nd law of Thermodynamics, an irreversible process always increases the entropy of the universe. Entropy is essentially a measure of disorder in the universe, and it makes sense that if you go through a process from beginning to end normally, the orderedness of the universe decreases.
Just remember the next time you're in physics class: ideal processes don't really exist!