I read somewhere that stable law is the special case of infinitely divisible. In other word, stable distribution is a special case of infinitely divisible distribution.
But I am not quite sure what differentiate both of them apart.
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Every stable law is infinitely divisible, but some infinitely divisible laws are not stable. An example of the latter is the Poisson distribution. For each $\lambda > 0$ and each $n\in\{1,2,3,\ldots\}$, there exist independent random variables $X_1,\ldots,X_n$ each distributed as $\mathrm{Poisson}(\lambda/n)$, so that $X_1+\cdots+X_n\sim\mathrm{Poisson}(\lambda)$. Thus each Poisson distribution is infinitely divisible. But the Poisson distribution is not stable, since, if you add two independent identically distributed random variables $X$ and $Y$, each with a Poisson distribution, what you get does not have the same distribution as $\alpha X+\beta$ for some $\alpha,\beta\in\mathbb R$.
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If a random variable $X$ is infinitely divisible, then for each $n$ we can write $$X = Y_1 + \cdots + Y_n$$ for some i.i.d. random variables $Y_i$. (Infinitely is a bit of a misnomer here -- all we really mean is that $n$ can be arbitrarily large.)
If $X$ is stable, then we can moreover do this in such a way that the $Y_i$ have the same distribution as $X$, up to scaling.
Daniel McLaury
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It seems like Poisson distribution would satisfy this definition, i.e. $X_1 +···+ X_n = c_n X +d_n$, – Liam Nousa Mar 29 '16 at 05:07