The word "representational," when used to describe a work of art, means that the work depicts something easily recognized by most people. Throughout our history as art-creating humans, most art has been representational. Even when art was symbolic, or non-figurative, it was usually representative of something. Abstract (non-representational) art is a relatively recent invention and didn't evolve until the early 20th-century.
There are three basic types of art: representational, abstract, and non-objective. Representational is the oldest, best-known, and most popular of the three.
Abstract art typically starts with a subject that exists in the real world but then presents those subjects in a new way. A well-known example of abstract art is Picasso's Three Musicians. Anyone looking at the painting would understand that its subjects are three individuals with musical instruments–but neither the musicians nor their instruments are intended to replicate reality.
Non-objective art does not, in any way, replicate or represent reality. Instead, it explores color, texture, and other visual elements without reference to natural or constructed world. Jackson Pollock, whose work involved complex splatters of paint, is a good example of a non-objective artist.
Representational art strives to depict reality. Because representational artists are creative individuals, however, their work need not look precisely like the object they are representing. For example, Impressionist artists such as Renoir and Monet used patches of color to create visually compelling, representative paintings of gardens, people, and locations.
Representational art got its start many millennia ago with Late Paleolithic figurines and carvings. Venus of Willendorf, while not too terribly realistic, is clearly meant to show the figure of a woman. She was created around 25,000 years ago and is an excellent example of the earliest representational art.
Ancient examples of representational art are often in the form of sculptures, decorative friezes, bas-reliefs, and busts representing real people, idealized gods, and scenes from nature. During the middle ages, European artists focused largely on religious subjects.
During the Renaissance, major artists such as Michaelangelo and Leonardo Da Vinci created extraordinarily realistic paintings and sculptures. Artists were also commissioned to paint portraits of members of the nobility. Some artists created workshops in which they trained apprentices in their own style of painting.
By the 19th century, representative artists were beginning to experiment with new ways of expressing themselves visually. They were also exploring new subjects: instead of focusing on portraits, landscapes, and religious subjects, artists experiments with socially relevant topics related to the Industrial Revolution.
Representational art is thriving. Many people have a higher degree of comfort with representational art than with abstract or non-objective art. Digital tools are providing artists with a wider range of options for capturing and creating realistic images.
Additionally, the workshop (or atelier) system continues to exist, and many of these teach figurative painting exclusively. One example is the School of Representational Art in Chicago, Illinois. There are also whole societies dedicated to representational art. Here in the United States, the Traditional Fine Arts Organization comes quickly to mind. A web search using the keywords of "representational + art + (your geographical location)" should turn up venues and/or artists in your area.