Background independence is a condition in theoretical physics, especially in quantum gravity (QG), 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, and in particular to not refer to a specific coordinate system or metric. The different configurations (or backgrounds) should be obtained as different solutions of the underlying equations.
What is background-independence? 
Background-independence is a loosely defined property of a theory of physics. Roughly speaking, it limits the number of mathematical structures used to describe space and time that are put in place "by hand". Instead, these structures are the result of dynamical equations, such as Einstein field equations, so that one can determine from first principles what form they should take. Since the form of the metric determines the result of calculations, a theory with background independence is more predictive than a theory without it, since the theory requires fewer inputs to make its predictions. This is analogous to desiring fewer free parameters in a fundamental theory. So background-independence can be seen as extending the mathematical objects that should be predicted from theory to include not just the parameters, but also geometrical structures. Summarizing this, Rickles writes: "Background structures are contrasted with dynamical ones, and a background independent theory only possesses the latter type—obviously, background dependent theories are those possessing the former type in addition to the latter type.".
In general relativity, background-independence is identified with the property that the metric of space-time is the solution of a dynamical equation. In classical mechanics, this is not the case, the metric is fixed by the physicist to match experimental observations. This is undesirable, since the form of the metric impacts the physical predictions, but is not itself predicted by the theory.
Manifest background-independence 
This is primarily an aesthetic rather than a physical requirement. It is analogous to requiring in differential geometry that equations be written in a form that is independent of the choice of charts and coordinate embeddings. If a background-independent formalism is present, it can lead to simpler and more elegant equations. However there is no physical content in requiring that a theory be manifestly background-independent – for example, the equations of general relativity can be rewritten in local coordinates without affecting the physical implications.
Although making a property manifest is only aesthetic, it is a useful tool for making sure the theory actually has that property. For example, if a theory is written in a manifestly lorentz invariant way, one can check at every step to be sure that lorentz invariance is preserved. Making a property manifest also makes it clear whether or not the theory actually has that property. The inability to make classical mechanics manifestly lorentz invariant does not reflect a lack of imagination on the part of the theorist, but rather a physical feature of the theory. The same goes for making classical mechanics, or electromagnetism background independent.
Theories of quantum gravity 
Because of the speculative nature of quantum gravity research, there is much debate as to the correct implementation of background-independence. Ultimately, the answer is to be decided by experiment, but until experiments can probe quantum gravity phenomena, physicists have to settle for debate. Below is a brief summary of the two largest quantum gravity approaches.
Physicists have studied models of 3D quantum gravity, which is a much simpler problem than 4D quantum gravity (this is because in 3D, quantum gravity has no local degrees of freedom). In these models, there are non-zero transition amplitudes between two different topologies, or in other words, the topology changes. This leads physicists to believe that any consistent quantum theory of gravity should include topology change as a dynamical process.
String theory 
String theory is usually formulated with perturbation theory around a fixed background. While it is possible that the theory defined this way is background-invariant, if so it is not manifest. One attempt to formulate string theory in a manifestly background-independent fashion is string field theory, but little progress has been made in understanding it.
Another approach is the AdS/CFT duality, which is believed to provide a full, non-perturbative definition of string theory in spacetimes with anti-de Sitter asymptotics. If so, this could describe a kind of superselection sector of the putative full, background-independent theory. A full non-perturbative definition of the theory in arbitrary space-time backgrounds is still lacking.
Topology change is an established process in string theory.
Loop quantum gravity 
A very different approach to quantum gravity called loop quantum gravity has been claimed to be background-independent, at least in the sense that geometric quantities, such as area, are predicted without reference to a background metric. However, one could say that the physics of loop quantum gravity is only background-independent in a weak sense. This is because it requires a fixed choice of topology for the space-time, which could be seen as a background structure.
See also 
- General relativity
- String theory
- Causal dynamical triangulation
- Loop quantum gravity
- Quantum field theory
- D. Rickles, Who's Afraid of Background Independence?, p. 4
- John Baez, Higher-Dimensional Algebra and Planck-Scale Physics
- Hiroshi Ooguri, Partition Functions and Topology-Changing Amplitudes in the 3D Lattice Gravity of Ponzano and Regge 
- L. Smolin, The case for background independence, hep-th/0507235
- C. Rovelli et al., Background independence in a nutshell, Class.Quant.Grav. 22 (2005) 2971–2990, gr-qc/0408079
- Edward Witten, Quantum Background Independence In String Theory, hep-th/9306122.
- J. Stachel, The Meaning of General Covariance: The Hole Story, in J. Earman, A. Janis, G. Massey and N. Rescher (eds.), Philosophical Problems of the Internal and External Worlds: Essays on the Philosophy of Adolf Grünbaum, University of Pittsburgh Press 1993, ISBN 0-8229-3738-7, pp. 129–160.
- J. Stachel, Changes in the Concepts of Space and Time Brought About by Relativity, in C. C. Gould and R. S. Cohen (eds.), Artifacts, Representations and Social Practice; Kluwer Academic 1994, ISBN 0-7923-2481-1, pp. 141–162.
- E. Zahar, Einstein's Revolution: A Study in Heuristic, ISBN 0-8126-9066-4