A symmetry of an equation is any operation which maps solutions to solutions; thus a symmetry is the same concept as a transform, except that the transformed equation is the same as the old.
In principle there are an infinite-dimensional space of symmetries; in practice, however, one works only with the finite-dimensional component of symmetries which have a clean and explicit algebraic description. Indeed many symmetries are linear in nature. Note that completely integrable equations enjoy an explicit infinite-dimensional space of symmetries, formed by using any of the infinite number of conserved quantities as a Hamiltonian.
The space of all symmetries form a group.
Symmetries can either be continuous (involving one or more continuously varying parameters) or discrete. Continuous symmetries are intimately related to conservation laws via Noether's theorem. Discrete symmetries do not directly generate conserved quantities, but they can be used to conjugate existing conserved quantities to new ones.
List of symmetries
Note that any given equation will typically only enjoy a subset of the symmetries on this list.