Rivers form when flowing water from precipitation or springs flows over land and accumulates in existing low points in the topography, eroding it and forming a riverbed. Rivers are channels for moving water, which always seeks to flow downhill, and indeed, many rivers start from small channels in mountains. As it erodes the landscape, different aspects of the interaction between flowing water and the ground cause winding and branching.
Rivers are the generic name for any large flow of water on land. They can be of vastly varying sizes, but smaller flows are often referred to by the names stream or creek or some other alternative. Rivers can form wherever there is both a source of water and relatively low channels in the land for it to flow into, but established rivers are better delineated than generic dips in the land. Rivers have definite banks, below which the ground is eroded away in a recognizable pattern. River beds are often easily identifiable, even when completely dry.
That said, while fast-moving rivers often dig away any soil, leaving a bed of only smoothed rocks, slower-moving rivers often deposit that soil, creating muddy riverbeds and murky water. These are the most important features of rivers in terms of supporting their own ecosystems.