an infinitely divisible distribution
Sums of n iid variables, suitably rescaled, are the classical objects of probability — but what laws can appear as the distribution of a sum for every n at once? A distribution mu on the real line is called infinitely divisible if, for every integer n >= 1, there is a probability law mu_n such that mu is the n-fold convolution mu_n * mu_n * ... * mu_n. Equivalently, a random variable X with law mu can, for each n, be written as a sum X = Y_1 + ... + Y_n of n iid pieces. This is exactly the class of laws that can sit at a fixed time of a process built from stationary independent increments, so infinite divisibility is the distributional gateway into Lévy processes.
In terms of the characteristic function phi(theta) = E[e^(i theta X)], infinite divisibility says phi = (phi_n)^n for each n, where phi_n is itself a characteristic function. The decisive structural fact is the Lévy-Khintchine formula: mu is infinitely divisible if and only if phi(theta) = e^(psi(theta)) with psi(theta) = i b theta - (1/2) sigma^2 theta^2 + integral over R\{0} of (e^(i theta x) - 1 - i theta x 1_{|x|<1}) nu(dx), where b is real, sigma^2 >= 0, and nu (the Lévy measure) satisfies integral of min(1, x^2) nu(dx) < infinity. Thus every ID law is a drift plus a Gaussian part plus a jump part, and these three ingredients are unique.
The normal, Poisson, gamma, Cauchy, all stable laws, the compound Poisson laws, and the inverse-Gaussian law are infinitely divisible; the uniform and the binomial are not (a bounded non-constant law cannot be divided indefinitely, and a binomial's atoms cannot be split). A common misconception is that infinite divisibility is a tail or moment condition — it is not; it is a convolution-algebraic condition, and an ID law can have very heavy tails (Cauchy) or all moments (normal).
The N(m, s^2) law is infinitely divisible: it equals the n-fold convolution of N(m/n, s^2/n). The Poisson(lambda) law equals the n-fold convolution of Poisson(lambda/n). But a fair coin's Bernoulli(1/2) law is not: if it were a sum of two iid pieces each could take at most two values, forcing a binomial which has three atoms, a contradiction.
Normal and Poisson divide forever; Bernoulli does not.
Infinite divisibility of the one-dimensional marginal is necessary AND sufficient for the law to be that of a Lévy process at time 1, but it says nothing on its own about path properties — those come from the Lévy measure via the Lévy-Itô decomposition.