Usually denoted as $f(x)$ or $f_X(x)$ for the random variable X.
The probability a variable will be between x = a and x = b is
$$P(a \leq x \leq b) = \int_{a}^b f(x) dx$$
A CDF, F(x) or $F_X(x)$, for the random variable X is given by:
$$F_X(x) = P(X \leq x),$$
where $P(X < x)$ is the probability that X is less than the value x.
A CDF may be found from a PDF by:
$$F_X(x) = \int_{-\infty}^x f_X(t) dt$$
The related function,
$$\bar{F} = 1 - F(x),$$
is called the complementary cumulative distribution function, tail distribution, or survival function.
The inverse of the CDF, $F^{-1}(x)$, called the quantile function.
Given the PDF, $f_X(x)$, for random variable X, to change variables to Y = g(X) (to get the PDF $f_Y(y)$:
$$f_Y(y) = \left\| \frac{d}{dx}(g^{-1}(y)) \right\| f_X(g^{-1}(y))$$
This comes from a relation between CDFs:
$$F_Y(y) = P(Y \leq y) = P(g(X) \leq y) = P(X \leq g^{-1}(y)) = F_X(g^{-1}(y))$$
The CF for a random variable X is defined as inverse Fourier transform of the PDF:
$$\phi_X(t) = \int_{-\infty}^{\infty} e^{itx} f_X(x) dx$$
The CF for a linear function of independent random variables,
$$Y = \sum a_i X_i$$,
is
$$\phi_Y = \prod \phi_{X_i}(a_i t)$$
allowing one to find the PDF of linear functions of variables by going to the CF domain and back.