When there is a circle around the integral signs that means that $S$ is a closed surface. In fact a surface integral is a special case of integrating what is called a differential form. In the case of a surface integral we are integrating what is known as a 2-form: for a line integral we integrate a 1-form and in the case of an ordinary integral of a function we are integrating a 0-form. Indeed there are higher forms that can be integrated such as in volume integrals (integrating a 3-form). Now to directly answer your question: yes surface integrals always sum differential area elements.