recovering the pmf or pdf from the cdf
The cdf accumulates probability; the pmf and pdf describe how probability is spread out. Since these are two views of the same thing, you can travel both ways. Going forward, you build the cdf by piling up mass or area. Going backward, you take the cdf apart again to read off the local description. This round trip is one of the most useful skills with distributions, because problems often hand you one form when you want the other.
For a discrete variable, the cdf is a staircase, and each jump corresponds to a value carrying probability. So the pmf is the size of the jump: p(x) = F(x) - F(x-), where F(x-) is the value just to the left (the limit approaching x from below). In words, the probability sitting exactly at x is how much the cdf leaps up there. Between jumps the cdf is flat, meaning no probability lives there. So you read a staircase by measuring the height of each step.
For a continuous variable, the cdf is smooth, and the rate at which it climbs is precisely the density. So the pdf is the derivative of the cdf: f(x) = F'(x), wherever that derivative exists. This makes sense because F(x) = integral from -infinity to x of f(t) dt, and by the fundamental theorem of calculus, differentiating an accumulated area gives back the integrand. Honest caveat: at the corners or jumps of a cdf the derivative may not exist, and a mixed distribution needs both moves at once, jumps for its lumps and a derivative for its smooth part.
If F(x) = 1 - e^(-x) for x >= 0 (and 0 before), differentiate to recover the density f(x) = e^(-x), the exponential pdf. If instead F jumps from 0.3 to 0.7 at x = 5, then p(5) = 0.7 - 0.3 = 0.4 sits exactly on the value 5.
Differentiate a smooth cdf to get the pdf; measure a jump in a staircase cdf to get the pmf.
A continuous cdf has no jumps, so P(X = x) = 0 everywhere; a pure staircase cdf has no slope between jumps, so it has no density. Mixed distributions are exactly those whose cdf both jumps and climbs.