An invariant is something that does not change under a set of transformations. The property of being an invariant is invariance. For the laymen, let us just say an invariant is some kind of correspondence between two types of mathematical objects, so that two 'similar' things correspond to one and the same object. Invariants are useful in discriminating complicated objects.
Mathematicians say that a quantity is invariant "under" a transformation; some economists say it is invariant "to" a transformation.
Some examples, taking more complicated objects to numbers:
The degree of a polynomial, under linear change of variables
The dimension of a topological object, under homeomorphism
In physics, invariants are usually quantities conserved (unchanged) by the symmetries of the physical system. (See Noether's theorem.) The correspondance between symmetries and conserved quantities is apparent through conservation laws. Much work has been done, esp. in quantum physics, to relate every conserved quantity to some symmetry. One such quantity that still defies all such attempts is mass. Some other examples include:
In computer science, optimising compilers and the methodology of design by contract pay close attention to invariant quantities in computer programs, where the set of transformations involved is the execution of the steps of the computer program.
A loop invariant is a constraint that should be satisfied by every iteration of a loop.
A class invariant is a constraint that should be satisfied by every method of a class.
In music using the twelve tone technique invariance describes the portions of rowss which have been so designed that they remain invariant under the allowable transformations (inversion, retrograde, retrograde-inversion, multiplication). George Perle describes their use as "pivots" or non-tonal ways of emphasizing certain pitcheses. Invariant rows are also combinatorial.
