In three dimensions we can integrate over a surface that is sufficiently nice, by dividing it into small pieces and proceeding exactly as for an area in the plane.
A plane is, after all, only a particularly simple and straight example of a surface in three dimensions.
We consider surfaces that for the most part look like planes at small distances so that each tiny piece of surface will have an area dS that is essentially a small piece of a plane.
However, each small piece of surface area, dS, has a normal direction n and once again it is appropriate to consider the vector dS which is its area, dS times its (outward) normal vector n.
You may sum all of these small vectors multiplied by an integrand and define Riemann sums for these to get a surface integral.
The last sentence above refers to multiplying the vector dS (remember dS is |dS| multiplied by the unit outward normal to the surface, n) by the integrand; and there are three obvious ways to do this.
If the integrand is a function f(x, y, z) we can multiply, and the sum will be a vector. This is OK but it is the least common thing to do.
The standard thing to do is to have an integrand vector v(x, y), and take its dot product with dS, and sum that dot product over the pieces. This is the most common form of surface integral.
We denote such an integral by
This kind of integral is particular useful in physical applications.
In particular when the vector v is a current density, vn is then defined to be the amount of whatever v is current density of that flows through a surface with normal n and surface area dS per unit time.
The integral above then tells how much of that stuff flows through the surface S per unit time.
Current densities are defined for mass, and charge, but surface integrals of this kind are important as well in discussions of electric and magnetic fields.
Gauss's Law, for example states that in electrostatics, the total electric charge within a region R is a constant times the integral over the surface R of that region of the component of the electric field normal to the surface
(There is a similar relation between the gravitational field and the amount of mass within a region; for magnetic fields, the apparent absence of magnetic charge (monopoles) means that the right hand side of the comparable equation is 0 for magnetic fields.)
Integrals of this kind are usually called Flux integrals.