Discrete Data
Discrete data consists of countable values with no values possible in between, such as the number of students in a class.
Definition
Discrete data is made up of separate, countable values with gaps between them: you can list the possible values, and there is no such thing as an "in between" value. It takes on a countable number of distinct values, often integers, and is visualized with bar graphs or dot plots rather than histograms with continuous bars. Formally, a discrete random variable $X$ takes values in a countable set $\{x_1, x_2, \ldots\}$ with probability mass function $p(x_i) = P(X=x_i)$, where $\sum p(x_i) = 1$; key discrete distributions include Binomial $B(n,p)$, Poisson($\lambda$), Geometric($p$), and Negative Binomial, and generating functions (PGF, MGF) facilitate computation of moments.
Example
The number of pets a family has ($0$, $1$, $2$, $3, \ldots$) is discrete, since you cannot have $1.7$ pets, and the number of goals scored in a soccer game is also discrete. The number of customers entering a store each hour ($0$, $1$, $2, \ldots$) is discrete too, appropriately modeled with Poisson (for counts) or binomial (for success/failure counts) distributions. A Poisson($\lambda=3$) model for customer arrivals gives $P(X=k) = e^{-3} \cdot 3^k / k!$, where the mean and variance are both $3$, a hallmark of the Poisson distribution.
Key Insight
If you can count it and there is no such thing as an in-between value, it is discrete; counting words in a sentence gives discrete data. Discrete distributions use probability mass functions (PMFs), and the sum of all PMF values equals $1$. The Poisson distribution arises as the limit of Binomial($n,p$) when $n$ is large and $p$ is small with $np = \lambda$ fixed, connecting the two most common count models.