How can a particle in a medium ever travel faster than light? In some media, mass-carrying particles can go faster than light:

Cherenkov radiation, ... is electromagnetic radiation emitted when a charged particle... passes through a dielectric medium at a speed greater than the phase velocity of light in that medium

How does this happen? I would imagine that light faces the least resistance to motion and that other charged mass-carriers have more stuff to impede their speed.
Why are the particles allowed to exceed the speed of light for other media when they are not allowed to in a vacuum?
 A: Cherenkov radiation is generated by charged particles which are faster than the local speed of light in an optical medium, which is always lower than the speed of light in vacuum. So if the local speed of light in an optical medium is given by c_medium = c_vacuum/n, where n is the refractive index, a particle with c>v>c_medium can generate Cherenkov radiation, even though it is slower than the actual speed of light in vacuum. Such particles can be found in cosmic rays and are made copiously in our accelerator facilities, where Cherenkov detectors are used often to detect them. The optical media used in Cherenkov detectors do slow particles down, of course. The Cherenkov light requires energy, which has to come out of the kinetic energy of these relativistic particles. Typically, however, particles can travel several meters, and sometimes many miles trough Cherenkov media before they have lost all of their energy. 
A: 
Why are the particles allowed to exceed the speed of light for other media when they are not allowed to in a vacuum?

Light is represented by electromagnetic radiation which in its simpler form is mathematically described by a sine or cosine wave, classically. The classical light is built up by photons which are the quantum of electromagnetic radiation and are the mediators of any interaction between charged particles. A charged particle entering a medium immediately starts interacting with the field of the atoms and molecules in the medium by exchanges of virtual photons, which we should keep in mind.
A sine or cosine wave mathematical description is not a good description for complex electromagnetic interactions . What is a good representation for an electromagnetic interaction  is a wave packet, i.e. combination of sines and cosines which can carry information by the phase and amplitude about the particle  passing through
The wave packet allows a mathematical description for an information carrying bundle (particle passing at x,y,z,t)  with sines and cosines but two definitions of velocity are needed, the group velocity, which is the peak of the bundle moving in space, 

and the phase velocity , which is the velocity of the individual sine wave component comprising the wave packet/bundle, as it is made up by a function that expands into  sine and cosine functions.
In a medium there is dispersion , and phase velocity , (i.e. how fast the sine and cosines comprising the wave packet are moving) can be different from the group velocity.
It is the group velocity in the medium that measures the velocity of light in the medium, i.e. how fast the information is given to the medium that a charged particle is passing is known to the medium with the group velocity: 
The wave packets of photons exchanged between the particle and the electromagnetic field of the atoms comprising the medium and say " charged particle passing" ,  are slower than the  velocity of  the particle, because of the index of refraction of the medium. The particle itself is not directly constrained by the index of refraction ( unless it is a photon) and can move faster than the group velocity of the electromagnetic wave packets that send the information that it is passing.
As the other answers say this can generate  Cerenkov radiation (analogous to sonic booms),  those exchanged wave packets stop being virtual and turn real, dissipating the energy from the wave packet. 
For electromagnetic wavepackets in vacuum the two velocities are the same because there is no dispersion .
A: 
Why are the particles allowed to exceed the speed of light for other
  media when they are not allowed to in a vacuum?

No particle can have relative speed greater than c, period.  This is 'built-in' to the geometry of spacetime.
But, a particle can travel faster, relative to some medium, than electromagnetic waves propagate in the same medium.
Do you see the distinction?  Such a particle isn't travelling faster than c relative to the medium.
If you see this, then your question may be why material particles can have a speed relative to the medium faster than light propagates in the medium.  
Consider the neutrino which is exceedingly 'reluctant' to interact with matter whereas photons readily interact with charged particles.  See, for example, this answer. 
