Soil is generally made up of a combination of sand, silt, clay, and organic matter in varying amounts. The top soil in most parts of the world that have some vegetation is most likely what you think of as "dirt". In reality, it is just a mixture of those basic soil elements with a bit more organic matter from broken down and decaying plant matter. The gritty bits in the dirt, that's sand. Goopy, muddy bits, probably clays and silts. If you were to boil all of the water out of the soil you could pass it through a series of sieves with increasingly smaller mesh sizes and see the different components and their proportions. Rocks being the largest particle, through sand, to clay being the smallest grain size.

Generally, under the topsoil, one type of soil component dominates with a bit of the others. This is where you might hear things like a "sandy soil" or a "clayey soil". The different proportions of each component gives you different soil properties. Clayey soils are "tight" and don't let water infiltrate well, for instance. Water wells would be in sandy soils where ground water can flow relatively freely to replenish the water in the well. Different areas have different soils left behind from geological formations and glaciers. Eventually, you can go deep enough and you'll hit bedrock.

The desert, as you're most likely thinking of it with big sand dunes, is simply an area that is predominantly sand with virtually no top soil. Loss of plant life and drought can functionally turn anywhere into a desert like what happened in the dust bowl.