These are variables that can take on a countable set of discrete values. The sum of the probabilities of each possible outcome is always $1$.
Expected value
The expected value (the mean) of a discrete random variable is given by the formula $$ \begin{equation}\begin{aligned} E(X)&=\sum{xP(X=x)}\\ \end{aligned}\end{equation} $$
where $X$ is the variable e.g. the face obtained from rolling a fair die and $x$ is a particular outcome e.g. $1$, $2$, etc.
Variance
The variance of the discrete random variable is $$ \begin{equation}\begin{aligned} Var(X)&=\sum{x^2P(X=x)}-[E(X)]^2\\ \end{aligned}\end{equation} $$
Transformations
For the transformation of a discrete random variable $X$, we can consider the mean/expected value: $$ \begin{equation}\begin{aligned} E(2X)&=2E(X)\\ E(2X+1)&=2E(X)+1\\ E(aX)&=aE(X)\\ E(aX+b)&=aE(X)+b\\ \end{aligned}\end{equation} $$
The variance for a transformation of $X$ is given by:
$$ \begin{equation}\begin{aligned} Var(2X)&=2^2Var(X)\\ Var(2X+1)&=2^2Var(X)\\ Var(aX)&=a^2Var(X)\\ Var(aX+b)&=a^2Var(X)\\ \end{aligned}\end{equation} $$Notice how the variance is not affected by the translation $b$. This is because variance is a measure of spread and translations do not affect how far apart the data points are (relative to each other) but rather only their positions relative to a standard value e.g. the value zero value of the scale.