Ohm's law

Law states that Voltage and Current are directly proportional, and the proportionality constant is collect R
V=R×I$V=R\times I$

Voltage is also an archaic word and concept to indicate a potential difference (also referred to as V or ΔV), that is the same thing that is related to energy. In other words it is the "Energy Gain" of a charge if it moves between the two extreme of the potential/voltage difference (if in a circuit there is 1 Volt difference from a to b, that is = to a gain of 1 Joule for 1 Coulomb of charge moved from a to b), and thus to very general and powerful physical concept.
Current is the flow of electric charge and is a way to count the charge that cross an hypothetical start-line boundary every second (1 Ampere is 1 Coulomb crossing the start line in 1 second).

In other word, Ohm's Law states that the bigger Energy gain the little electron fellas will have the more they will move. How much more? That's up to the resistance.
If the resistance is small than is like going in vacation following a large highway: a lot of charge will move very fast even if the energy gain is little.
If the resistance is big, than is like going in vacation following a small road: not many charges will move not very fast even if the energy gain is consistent.

That's because electrons have to transfer their energy (classical model say they move and bounce around, but is not exactly true), and the easiness of this happening is given by resistance, and if the material is not so suitable (very movable) than they will need a larger incentive for doing that.

Similar to highways, the larger they are the easier the people crowd them, and the longer they have to be followed, more motivated the people have to be for going all the way.

So here it comes the concept of resistivity:
resistivity is the intrisic properties of a material to allow or enable the energy transfer that let the current flow. 
Then the larger the cable, the easier the current flow, and the longer is the cable the harder will be for the charge carrier, and so the resistance is given by
R=ρ×l/A$R=\rho \times l/A$
where ρ$\rho$ is the resistivity, l the length, and A the section of the resistance.

Conductivity is simply the inverse of it.

Most often this behavior is well followed, even if, as all Physics, have it's limits of applicability it's very general that a macroscopic chunk of uniform material will, at least in first approximation, follow the Ohm Law up to very exotic stretching in very special cases.

The thing "in first approximation" imply that every material have its degree of "non-ohmic behaviour" even in the most standard regime. For example if you take a resistor and a current flowing in it, the resistor will heat up. This imply that its molecular properties and geometrical configuration, at least by a little, shaken up, and so the resistivity and the resistance will, at least by a little, change.

But in general, "non ohmic materials" are more of a thing that goes into 

• - microscopic world: few-atoms systems or crystalline configurations
• - properties of composite devices: semiconductors devices like transistors and diodes that are built specifically to have peculiar electric response
• - materials under extreme conditions: the Tungsten inside an incandescence lamp is so heated up that emit light, obviously is geometrical and atomic configuration are completely messed up and its far over the regime of ohmic behavior

and so the range of applicability of Ohm Law is surprisingly vast (really, for being a thing devised without the faintest idea of what atomic structure, charge carriers and electromagnetic force are holds up amazingly!).
And a Law is simply that: a general description of a general behavior that have a vast range of applicability and holds up to the test of time.

PS: As a bonus I flying cite the fact that Ohm's Law is Aristotical (Energy proportional to velocity) while almost all other laws in physics are Newtonian (Energy proportional to velocity squared). Maybe there is even more to Ohm's law that we are still not able to see!