I'll expand a bit on what Frank has said. These hydrogen orbitals have been found as solutions to Schrodinger's equation for a single electron bound by the potential well of the Coulomb force exerted by the proton on the electron. When this LINEAR differential equation is solved, we find that there are solutions only for discrete energy values, which are eigenvalues of the linear system. (This is true as long as the electron is bound, meaning it has insufficient energy to escape.) These are denoted by the energy quantum number.
Next, these eigenvalues are used to solve for the eigenfunctions associated with them. These are the associated wave functions. Here we arrive at the first orthogonality condition. Eigenfunctions associated with different eigenvalues are orthogonal.
For the ground state ( n = 1 ) we found just one eigenfunction associated with the energy level. That is the 1S orbital. (Which can be occupied by 2 electrons of opposite spin, but that goes a bit farther.) But for n=2, we find 4 eigenfunctions, all having the same energy level: 2S, 2Px, 2Py, 2Pz. (There are other considerations in multi-electron atoms, which lower the S orbital energies compared to the P orbitals.) So here we have a second case of orthogonal functions. We have CHOSEN to construct orthogonal wavefunctions for mathematical convenience, For this case, if we consider a hydrogen atom in the n=2 energy state, the atom would have that energy in any of these states. But, more than that, these functions are the solutions to a linear differential equation. This means that any linear combination is also a solution. This set of orbitals plays the same role in forming the wavefunction that is played by the traditional i, j, k unit vectors to describe a vector in 3-space. We don't require every vector to be either i or j or k. Rather we write any vector as xi+yj+zk. In the same way, any linear combination of 2S, 2Px, 2Py, 2Pz orbital is a legitimate wavefunction for the n=2 quantum state, so long it is normalized so that the Integral of Y* Y over all space is equal to 1. (Since there is exactly probability=1 of finding the electron somewhere.)
In fact, when considering these orbitals being involved in bonding between atoms to form a molecule, it is often the case that some linear combination offers a better bond -- meaning lower energy. Then we see things like SP hybrid orbitals involved in bonding.
