The Iris dataset is deservedly widely used throughout statistical science, especially for illustrating various problems in statistical graphics, multivariate statistics and machine learning.
Containing 150 observations, it is small but not trivial.
The task it poses of discriminating between three species of Iris from measurements of their petals and sepals is simple but challenging.
The data are real data, but apparently of good quality. In principle and in practice, test datasets could be synthetic and that might be necessary or useful to make a point. Nevertheless, few people object to real data.
The data were used by the celebrated British statistician Ronald Fisher in 1936. (Later he was knighted and became Sir Ronald.) At least some teachers like the idea of a dataset with a link to someone so well known within the field. The data were originally published by the statistically-minded botanist Edgar Anderson, but that earlier origin does not diminish the association.
Using a few famous datasets is one of the traditions we hand down, such as telling each new generation that Student worked for Guinness or that many famous statisticians fell out with each other. That may sound like inertia, but in comparing methods old and new, and in evaluating any method, it is often considered helpful to try them out on known datasets, thus maintaining some continuity in how we assess methods.
Last, but not least, the Iris dataset can be enjoyably coupled with pictures of the flowers concerned, as from e.g. the useful Wikipedia entry on the dataset.
Note. Do your bit for biological correctness in citing the plants concerned carefully. Iris setosa, Iris versicolor and Iris virginica are three species (not varieties, as in some statistical accounts); their binominals should be presented in italic, as here; and Iris as genus name and the other names indicating particular species should begin with upper and lower case respectively.
(EDIT 4 May 2022 In a generally excellent book to hand on machine learning, the Iris data are described in terms of classes, types, kinds and subspecies, but never once correctly from a biological viewpoint. Naturally that sloppiness makes not a jot of difference to the machine learning exposition.)
Stebbins (1978) gave an appreciation of Anderson, a distinguished and
idiosyncratic botanist, and comments on the scientific background to
distinguishing three species of the genus Iris. Kleinman (2002)
surveys Anderson's graphical contributions with statistical flavor. See also Unwin and Kleinman (2021).
Kleinman, K. 2002.
How graphical innovations assisted Edgar Anderson's discoveries in
Chance 15(3): 17-21.
Stebbins, G. L. 1978. Edgar Anderson 1897--1969.
Biographical Memoir. Washington, DC: National Academy of Sciences.
Unwin, A. and Kleinman, K. 2021. The iris data set: In search of the source of virginica. Significance 18: 26-29. https://doi.org/10.1111/1740-9713.01589