History
To extend a bit on what was said:
Schrödinger, according to Dirac, wrote down the relativistic equation for a spinless particle first. He arrived at this by noticing the experimental relationship between the energy of a particle (like an electron or a photon) and the frequency of the wave-behavior of that particle, and similarly between the momentum and the wavelength. For example, Einstein found that individual photons carried an energy E=hv, where v is the frequency of light used. Imposing the relativistic relationship
E^2=p^2+m^2 (with c=1), he arrived at the relativistic equation that describes a function that obeys a wave equation, so that this function would describe the state of a particle with mass m with "wavelike" properties as experimentally observed.
However, Schrödinger noticed that the while the equation agreed with the "rough" spectrum of Hydrogen, it did not agree with the "fine structure" corrections. He decided that the non-relativistic form of his equation would at least give an agreement with the rough spectrum of Hydrogen, so it was capturing something right...the relativistic form would go too far and make false predictions.
The resolution came with the realization that there must be an intrinsic angular momentum associated with the electron in the bound state with the proton making up hydrogen. The corrections to the energy spectrum of hydrogen (using the original *relativistic* form of Schrodingers equation) due to spin effects gave the right results for the fine structure.
Dirac had a problem, though, with the relativistic equation, and the same problem existed with his own equation (which took into acount the spin of the electron from the start). The problem is that the Hamiltonian would be unbounded below...this meant that electrons would seek lower energy configurations by tunneling to negative energy states and keep going down in energy, never reaching a stable energy configuration.
A resolution to this is that there are antiparticles, particles with opposite "quantum numbers", and that (e.g.) an electron can't spontaneously become an anti-electron. Therefore, Dirac's equation described two types of particles with energies bounded below by mc^2.
However, as experiments became more sensitive, it was found that there were further splittings that the new relativistic quantum mechanics didn't explain. The resolution was QED (and more generally, quantum fields): there were "radiative corrections" to the spectrum of Hydrogen due to the interaction of the electron with the quantum "Maxwell" field. That is, an electron near a proton interacts with discrete bits of a field, which are photons, on top of the classical electromagnetic background field (which can, in turn, be seen as a bunch of photons in a "coherent quantum state"). The interaction with these photons is in a quantum fashion, meaning the effect is, in a specific sense, due to all numbers of photons interacting in all possible ways with the electron, the overall effect being that the electron feels some effective fuzzy influence of it. Another example of this fuzzy influence that is more relevant to what you see in chemistry, e.g., is the state of a single electron in a simple covalent molecule: the electron is delocalized around the two nuclei (being in all places allowed at once) forming a glue for the molecule (if the electron didn't delocalize in this fahion, the molecule would not be stable).
By the way, from a modern perspective, quantum field theory is the necessary way to combine quantum mechanics and special relativity...the need to use fields arises naturally when you try to do this. Also, antiparticles are an immediate consequence in qft when you simply demand conservation of a set of quantum numbers (like electric charge).
