All irreversible (natural and spontaneous) processes are characterized by the fact that entropy always increases in such processes.
And the second law of thermodynamics logically means that entropy always tends to increase.
A physical system shall always proceed to a state of maximum entropy.
In other words, second law specifies the direction of evolution of a natural process.
Natural systems always have a tendency to maximise their entropy.
And that's what the second law is all about.
Consider for example, the transfer of heat from one body to another in contact due to the temperature difference.
Heat always flows from a hotter body to a colder one spontaneously. But, no one has ever observed spontaneous transfer of heat from a colder body to a hotter body.
Even though such phenomenon are permissible by the first law, such processes never occur naturally. That's the essence of the second law.
Heat is transferred from a hotter body to a colder body because it is accompanied by entropy increase but, the converse never occurs because as such the entropy of the system is required to decrease.
That's what the Clausius' statement is all about.
It may be proved that all statements of the second law are completely equivalent and revolves around the same central concept of entropy increase.
It may be noted that, the transfer of heat from a colder body to a hotter body is possible (as in a refrigerator or an air conditioner). Second law states that such a process is however not spontaneous and natural. In order for such a process to take place, external work is required.