Why do some conductors follow Ohm's law and some do not? Isn't there any universal law that can explain the flow of current?
Physics does not contain any truly 'universal' laws yet. Ohm's law like any other within physics relies on context for validity. $V=IR$ is implicitly only valid in simple circuit theory, in field theory it has the form $J=\sigma E$ as seen in the wiki article.
Variations from Ohm's law are due to electrical properties of the conductor/medium under inspection. The 'law' or equation to be used will be determined by pragmatic concerns of usefulness.
Ohm's law is an assumption that the value of resistance for a resistor is independent of the magnitude and polarity of an applied potential difference. Many conductors, like a diode, depend on the polarity of the applied potential difference-- e.g. you must apply a positive or negative voltage to get a current to move across the conductor.
An ohmic resistor on the other hand, will have a current that is linearly dependent on potential-- so as a larger positive voltage is applied to the resistor, there will be a correspondingly larger positive current through the resistor. Since R = V/I in a circuit, if both V and I are scaling up or down at a constant, linearly proportional rate, R will be constant since the factors that scale the numerator and denominator cancel each other out... and this is what it is meant by a resistor to be ohmic.
Most conductors change their resistance as their temperature change, and like a diodes, some conductors have completely non-linear relationships between current and voltage. Because Ohm's law isn't really a law, a more consistent way to look at a material is to consider it's "instantaneous" resistance, or it's resistivity (a property of a material as opposed to a geometry of a material):
$\rho$ is defined as the resistivity -- this is given by the electric field, $E$, divided by the current density $J$, (the current per cross-sectional area). So intuitively, a material with a high resistivity has a low current density motivated by a large electric field (remember, e-fields are proportional to the force on a particle), and a material with a low resistivity has a large current density motivated by a low electric field...
In any case, the salient thing to take away from this is that Ohm's law is only a "law" because of historical reasons-- in reality Ohm's law describes a special subgroup of conductors that have a linear relationship between current and voltage, but there are many conductors which don't fit this constraint.