I read this definition:

reversible process is one which can be stopped at any stage and reversed so that the system and surroundings are exactly restored to their initial states. An irreversible process is one in which heat is transferred through a finite temperature.

But I just don't get it yet. How can we really differentiate between an irreversible and a reversible process with only measurements or calculations? How do U, H, T, Q and P behave? I only clearly understand that in a reversible process maximum work is done, we are taking more steps to change from one state to the other and entropy must stay constant, the order of the particles of the system can't be restored once it changes, if we had 1 mole it would be hard for $6.022 \times 10^{23}$ atoms to be in the same state as before.

But it is not as clear, for me, how the other variables are affected if we have a reversible or irreversible process.