A renormalization-group fixed point is a set of couplings left unchanged by coarse-graining. It describes a scale-invariant theory.
An infrared fixed point attracts renormalization-group flows toward low energies or long distances. If it is interacting, it defines a nontrivial scale-invariant quantum or statistical field theory.
The stability matrix is the Jacobian of the beta functions at a renormalization-group fixed point. Its eigenvalues classify perturbations as relevant, irrelevant, or marginal.
The critical surface is the stable manifold of a critical renormalization-group fixed point. Tuning every relevant direction places a theory on this surface.

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