Symmetries are called "internal" to distinguish them from spatial symmetries.
The initial idea behind the use of symmetries was motivated by degeneracies in the spectrum of the Hamiltonian. If you have elements of some symmetry group $G=\{g_1,g_2,\ldots,\}$, and $g_iH=Hg_i$ for every element in the group, then all elements in $G$ will take an eigenstate of $H$ to another eigenstate of $H$ with the same energy. Those states form a multiplet.
This is already clear for the hydrogen atom. The energies $E_n=-13.6\text{eV}/n^2$ does not depend on the angular momentum $\ell$ of the system, and indeed there is a (continuous) group SO(3) so that any element commutes with $H$: this is obvious since $H$ is rotationally invariant. Note that even if elements in SO(3) are defined from $3\times 3$ orthogonal matrices, one can easily construct matrices of size $(2L+1)\times (2L+1)$ using the usual angular momentum operators $J_z, J_\pm$ that will commute with $H$. By exponentiating these matrices we obtain a set of $(2L+1)\times (2L+1)$ matrices that multiply in exactly the same way as the original $3\times 3$ orthogonal matrices.
There is a similar idea for the n-dimensional harmonic oscillator, but the symmetry group here is $U(n)$: see this question.
Crystals also have a variety of (discrete) spatial symmetries and indeed some of the first applications in physics of group theory were to understand the vibration of crystals. The pioneering work of Hans Bethe
H. Bethe, Splitting of terms in Crystals, Ann.Physik 3 (1929) 133-206
remains a classic reference.
Heisenberg noted that the proton and neutron have approximately the same mass and (neglectic the electric charge), appear to interact in the same way in the nucleus. Thus he postulated that a proton and a neutron were like the spin-up and spin-down states in systems where interactions did not depend on spin, i.e. he assumed that $SU(2)$ transformations operated on neutrons and protons just like they operate on spin-up and spin-down states, and so these $SU(2)$ transformations between neutrons and protons should commute with the nuclear Hamiltonian. This symmetry has nothing to do with actual physical space but is a symmetry in some abstract space where basis states are $\vert n\rangle\mapsto (1,0)^\top$ and $\vert p\rangle\mapsto (0,1)^\top$.
The idea of such an ``internal'' symmetry'' has since been generalized beyond $SU(2)$ to include various groups. The value of such internal symmetries is in how they connect predict various quantities (masses, cross-sections, etc.).