I agree with other answers and comments that your textbook does not have things stated correctly so I have explained below in words only what I see as the difference between Irreversible and Reversible processes. (You will find spontaneous reactions hidden in these statements)
Every system left to itself will change, rapidly or slowly, in such a way as to reach a state of rest defined in a statistical way and this is also called the state of equilibrium. The system will only move away from its state of equilibrium through the influence of some external events. We are familiar with many processes that reach equilibrium, diffusion of a concentrated solution into a dilute one leading to uniform concentration, transfer of heat from a hot to a cold body leading to uniform temperature, oxidation of substances by the atmosphere, self demagnetisation of magnets are all examples of spontaneous events in nature.
These processes and all other natural processes are similar in one respect, that they all bring the system to equilibrium, and we may think of these systems as loosing some measure of their capacity for spontaneous change.
A system far from equilibrium is one which we would choose to harness for doing useful work, for example the combustion of coal in air to cause a steam engine to operate. However, the second law shows us that it is not the loss of energy that is important but rather the availability of the energy for external purposes.
The essential content of the second law can be given by the statement that when any actual process occurs it is impossible to invent a means of restoring every system concerned to its original condition. Therefore, in a technical sense, any actual process is said to be irreversible.
The ideal or reversible process
We now distinguish between an actual system, which is always irreversible, and an ideal one, which although never occurring in nature is nonetheless imaginable. Such an ideal process is called reversible. In such a process each stage is conducted so that an infinitesimal change in the external conditions would cause a change in the direction of the process. In this sense such an imaginary system is called reversible.
An example is a system of water and water vapour inside a cylinder with a frictionless moveable piston. (In practice, by careful engineering, we can make the friction so small that it is negligible). Outside the piston is gas at some pressure. At constant temperature the system comes to equilibrium with respect to external conditions (pressure , temperature). If now the external pressure is increased, by the piston moving in by an infinitesimal amount, water vapour condenses. If the pressure is now reduced by an infinitesimal amount, some water evaporates. Thus the work required to vaporise 1 mole of water and to condense 1 mole of vapour differs only by an infinitesimal amount.
Another example is provided by an electrochemical cell where the cell potential can be measured extremely accurately with a voltmeter. The cell can be balanced so finely against an external emf that the current that can be made to flow in one direction or the other is only microamps.
A similar situation applies to thermal processes, since if two bodies differ in temperature by only an infinitesimal amount, the transfer of heat is likewise a reversible process for it would be possible to restore the system to its original condition with only an infinitesimal change in the external system.
(for a fuller description see Lewis & Randall 'Thermodynamics')