Mode
The mode is the value that appears most frequently in a dataset.
Definition
The mode is the value that appears the most in a set of data; a dataset can have one mode, two modes (bimodal), or no mode if all values appear equally, and for qualitative data it is the category with the highest frequency. A dataset may be unimodal, bimodal, or multimodal, and if all values have equal frequency there is no mode. Formally, the mode of a continuous distribution is the value $x^*$ that maximizes the probability density function ($f'(x^*) = 0$ and $f''(x^*) < 0$); for a sample, the sample mode is not a well-defined statistic for continuous data, so mode estimation for continuous distributions uses kernel density estimation, with the mode identified as the peak of the KDE curve.
Example
Shoe sizes in a class of $7$, $8$, $8$, $9$, $9$, $9$, $10$, $10$ have mode $9$ because it appears $3$ times, more than any other size. Categorical survey data on favorite genre, Action ($45$), Comedy ($30$), Drama ($45$), Horror ($20$), is bimodal since Action and Drama both appear $45$ times. A normal distribution $N(\mu, \sigma^2)$ is unimodal with mode $= \mu = \text{mean} = \text{median}$, a special property of symmetric distributions, while a beta distribution $\text{Beta}(\alpha, \beta)$ with both $\alpha > 1$ and $\beta > 1$ has mode $= (\alpha-1)/(\alpha+\beta-2)$.
Key Insight
The mode is the only measure of center that works for categories: you can find the most popular flavor, color, or answer, but you cannot average them, and it corresponds to the peak(s) of the frequency distribution. In Bayesian statistics, the mode of the posterior distribution is called the MAP (maximum a posteriori) estimate, which corresponds to penalized MLE and is a common point estimate when the posterior is intractable.