Second law of thermodynamics tells that:
Energy tends to disperse from being localized if it is not hindered from doing so.
Why does a hot cup of tea become cool? Because the higher vibrational energy of the hot mug & tea was localized only to the tea-mug system. When it is brought in the surroundings, the localized energy disperses to the surroundings. This is what second law of thermodynamics says about.
Let me give you another example of energy dispersal approach to entropy.
Suppose there is a gas enclosed by a stop-cock in of the bulbs of a two-bulb container. When you open the stop-cock, the gas expands & covers the two bulbs. Why did it happen so? It happened because as you open the stop-cock(removed the hinderance), the localized energy of the gas molecules spread over the whole volume to disperse its energy over a larger volume.
Change in entropy measures how much energy has been dispersed at a specific temperature. Entropy is a state property & how it changes doesn't matter at all. However it is far easy to take the course of the change as a reversible process.
Entropy of the universe(system + surroundings) always increases or doesn't change but never decreases.
For example, take a hot body at temperature $T$. Now, bring in another body which is at a temperature $T - dT$. Since, the difference is infinitesimal, the process of transfer of heat can be considered as reversible. Let $dQ$ heat energy be transferred from the hotter body to the colder body. Decrease in entropy of the hotter body after the heat transfer is $\dfrac{-dQ}{T}$ & increase in entropy of the colder body is $\dfrac{dQ}{T - dT}$. By simple inequality, we can show that $$\dfrac{dQ}{T - dT} > \dfrac{dQ}{T} \implies -\dfrac{dQ}{T - dT} < -\dfrac{dQ}{T} \implies \dfrac{dQ}{T - dT} - \dfrac{dQ}{T} > 0 \implies dS > 0$$.
In classical thermodynamics, there is no definition of entropy actually; what is concerned here is actually the change of entropy.
However, in statistical Mechanics, entropy can be defined as the number of microstates the system has at a particular state.
What is a microstate? each particle in a system collides & now & then they redistribute the energy among themselves without changing the total energy of the system. Microstate is one of the arrangements of the particle's energy for a particular state. It describes how each molecule is distributed at various energy levels at a certain instant. ( As you are reading Atkins, you can get the taste of this approach soon; he has described it beautifully.) It is not that the particles are at various microstates at a single time; it rather describes at a certain instant how the particles are distributed among various energy levels(Total energy is always same; it is just redistributed now & then due to the random collisions of the particles.)
More the microstates for a given state(temperature, pressure, internal energy of the system), more is the entropy at that state. Why? Because here the total energy has more ways to get dispersed to a certain microstate. Suppose, you have two systems $A$ & $B$ having the same energy $U$. However, $A$ has more microstates than $B$. So, the energy $U$ in $A$ owing to its having larger number of microstates has more ways to redistribute among the particles at a certain instant than in case of $B$. So, the energy $U$ disperses more in $A$( as it has more ways to distribute its internal energy) than in $B$ where the energy $U$ disperses lesser (since it has lesser number of ways to redistribute its energy) & hence its energy is localized. Thus entropy of $A$ is more in $A$ than in $B$ as in the former, energy isn't localized & disperses to more microstates than in $B$ .
