LED stands for light-emitting diode.
So an LED is a specific type of diode that when a forward bias "positive voltage" across the diode it emits light.
So you ask how does a diode work? (Here is the technical details from semiconductors class)
Background:
There are certain elements on this Earth which are called semiconductors. So now you ask what is a semiconductor? Okay a semiconductor is an element when in it's elemental form and at absolute zero is an insulator. As temperature increases it becomes more of a conductor (it is not due directly to temp itself, but I will explain below), not as good as a metallic substance. This property is due to the merging of electron valence shells in the crystalline form. When in crystalline form and at higher temps, the bands separate into different bands (groups of locations in which electrons can reside), noteably the valence and conduction bands. In a pure semiconductor crystal, electrons with a high enough energy will jump from the valence band into the conduction band. This means now the atom is minus one electron. This absence is actually sometimes considered a particle in itself but we just call it a hole. This hole gets passed around in the valence band like the electron gets passed around in the conduction band. This is current even if random, now this has made the crystal more of a conductor. Now I alluded to temp being a reason why the pure semiconductor becomes a conductor under certain cirumstances. Well what is temperature, the average kinetic energy of the atom. That should be self-explanatory.
More background:
Okay we talked about electrons and holes. Now the only problem is that the above doesn't happen enough in pure semiconductors to have the desired effect on the large scale. So how do we get the desired effect? Somehow we have to increase the density of electrons or holes in the material. So we dope (intentionally add impurities) the material by either adding materials with +1 or -1 more electrons in the atomic shell than the semiconductor. With the +1 atoms, they donate to the concentration of electrons, they are called donors. With the -1 atoms, they accept electrons, therefore they add to the hole concentration and they are called acceptors. A semiconductor which is doped with donors is called a n-type semiconductor (think of negative, it has more electrons than holes). A semiconductor which is doped with acceptors is called a p-type semiconductor (think of positive, has more holes than electrons).
Even more background:
This is the more complicated part on this path to learning how a diode works. The carrier actions: drift, diffusion, recombination-generation. Drift is random thermal action. A carrier goes in a direction until it is deflected (careful never to say hits or collides because they never touch they are repelled by electric fields) in a different direction. This leads to diffusion which depends heavily on drift. These carriers drift and disperse into areas with lower concentrations of the carrier by diffusion pressure. I don't want to explain all of the recombination-generation processes, but this is basically all the times that the carriers are generated or annihilated. A hole forms when an electron leaves the valence band and the carriers are annihilated when the electron comes back into the valence band.
Another thing to talk about and this leads into the main topic, that of the diode. If you have an unevenly doped semiconductor, the carriers will try to diffuse until the carriers are evenly distributed but it never will distribute evenly because it will be opposed by the force of an electric field and it will come to equilibrium without being evenly distributed.
Diodes:
Okay, now if you understood everything above, then you are pretty much ready to understand a detailed explanation of a diode. A diode is a semiconductor which has a p-type and a n-type semiconductor joined together. Remember, p-type has an excess of holes and n-type has an excess electrons, so they diffuse to the other side of the diode respectively. The local electric fields in the diode forms a depletion region around the junction where there are no carriers although the doped ions are still there. This depletion region we will call a potential hill, because that is what it is. Carriers can slide down the hill to their own side but the carrier already on their own slide can only go up the hill if they have that much potential energy. Okay, this is what is happening in a diode with no bias (no applied voltage). Now the applied voltage changes the size of the hill. A forward bias will lower the hill allowing current to flow exponentially with respect to voltage. A reverse bias will raise the hill effectively turning it into a resistor. Okay, now you know what a diode does, so what about the light-emitting diode. It is a diode made specific semiconductors to produce the desired form of electomagnetic radiation. They are all over the spectrum: microwave, infrared, visible, etc... So when forward biased, the electrons will give off their energy when recombined and this energy is related to wavelength by Planck's constant and that is the light you will get.
I hope that answered your question. Forgive any mistakes.