Definition
A scientific theory is a comprehensive explanation of some aspect of the natural world, built on accumulated evidence and successfully tested by repeated experiment and observation.
In ordinary speech "theory" often means "guess." In science it means almost the opposite: a well-supported framework that has survived attempts to falsify it and that makes further testable predictions.
Why it matters
How it works
Hawking's working definition is useful: a good theory must accurately describe a large class of observations on the basis of a model with few arbitrary elements, and it must make definite predictions about the results of future observations. The first requirement is parsimony; the second is testability.
A theory typically combines mathematical structure with empirical content. Newton's theory of gravitation, for instance, postulates a universal inverse-square force and derives planetary orbits, tides, and the trajectory of projectiles from a single equation with a single constant. The theory's success is measured by how many independent observations it explains and predicts — and by how few free parameters it needs.
The status of a theory depends on its track record. New theories begin as hypotheses, get tested, and if they survive enough tests with enough precision they earn the broader title. Even a "well-established" theory remains in principle revisable: Newton's gravity stood for over two centuries before Einstein replaced it. The replacement did not show Newton was "wrong" so much as that his theory was an excellent approximation in a restricted regime. This is the typical fate of successful theories — they get embedded as limits of more general successors. Quantum mechanics and general relativity are the current giants awaiting a theory that subsumes both.