Elements have more than one spectral line because electrons can occupy more than one excited state in the atom. Excited states correspond to different amounts of energy, so when the electron drops between different states, the energy given off corresponds to more than one spectral line.
Spectral lines come from the quantum mechanics of atoms. The amount of energy contained in the electrons in an atom determines where that electron orbits. Larger energies result in higher energy levels. Quantum mechanics is important because the energy levels of electrons in an atom are "quantized," meaning that the electron can only have certain energies. The energies are separated by integer multiples of the ground state energy.
Considering a hydrogen ion (one proton and one electron), the single electron can have several different energy levels, depending on how much energy is in the electron. Although the individual energy levels are the same distance apart, there is a different energy associated with dropping from the second to the first energy level or from the second all the way down to the ground state.
Emission spectra comes from a collection of many different atoms of an elemental gas. Therefore, in some atoms, the electron is dropping one energy level, while in other atoms, the electron is dropping two, three, or more energy levels. The result is that more than one spectral line shows up in the emission spectra.