Ohm's Law simply states that for a given resistor, the current and the voltage are directly proportional. So if you increase the voltage by 10%, the current goes up by 10%. If you halve the voltage, the current is halved.
This matches up quite well with the old water-in-a-pipe analogy that you often hear about when discussing electricity. If you take a section of pipe (resistor) and apply water pressure at one end (voltage), you will get a flow of water (current). If you increase the pressure, you get more flow, with a roughly proportional relationship - double the pressure, you get double the flow.
The reason Ohm's Law is so simple is that when electrons are flowing in a conductor, the underlying physical process is rather like water flowing in a pipe - electrons can move from atom to atom with some resistance that is (roughly) independent of the voltage.
However, Ohm's Law has its limits. One problem is that the current heats up the resistor. This increases the resistance, which means you don't get the exact current increase that Ohm's Law predicts.
Semi-conductors don't obey Ohm's Law because there is a different physical mechanism at play. The electrons, instead of shuffling past atoms that are in in a continuous lattice, have to jump between quantum energy levels.
Superconductors don't obey it either because again there is a different physical process - it involves Bose-Einstein condensates, which is straying a bit out-of-scope..
Ohm's Law should be called Ohm's Rough Guideline.