Historical note
The term “fractal”, from the Latin fractus (meaning “broken”), was proposed by Polish-born French mathematician Benoit Mandelbrot, around 1975, to refer to a 
concept that was developed to study irregular or fragmented processes and forms found in many structures in nature, such as the coastlines of land masses, sponges, cloud formations, holes in Swiss cheese, certain varieties of cabbage, etc.
Examples
Educational note
Although this aspect of geometry is not taught in pre-university math programs, it may be interesting to note that the results of studies in this particular area of geometry have had many practical applications throughout the years, such as in the development of the coffee percolation process, the formation of gels, the development of the vulcanization process, the optimization of throughput in communications networks, the study of how epidemics spread, etc.