Background independence is a condition in theoretical physics, especially in quantum gravity, that requires the defining equations of a theory to be independent of the actual shape of the spacetime and the value of various fields within the spacetime. The different configurations (or backgrounds) should be obtained as different solutions of the underlying equations.
Although physics of string theory can be showed to be background-independent - there exists one string theory only, and this theory has many solutions - the current formulations of this theory don't make this independence manifest because they usually require the physicists to start with a particular solution; i.e. a concrete background. A very different approach to quantum gravity called loop quantum gravity is, at least formally, manifestly background-independent. However physics of loop quantum gravity is not quite background-independent. For example, it requires us to choose a topology of the space that can't be changed. Also, loop quantum gravity seems to violate Lorentz invariance.
The classical background-independent approach to string theory is string field theory. Although string field theory has been useful to understand tachyon condensation, most string theorists believe that it will never be useful to understand non-perturbative physics of string theory.