What causes aurorae? I know that aurorae are caused by solar activity interacting with the Earth's magnetic field as charged particles are funneled down into the polar magnetic mirrors. 

On one hand, I know that as charged particles accelerate, they will produce radiation. They will happen in at least two ways in the context of aurorae: 


Charged particles will gyrate around magnetic field lines with a characteristic frequency, which presumably varies with pitch angle and position in Earth's magnetic field. 
As the charged particles are slowed down by the magnetic mirror effect, they will release some radiation. 


I've also heard that the visible radiation is caused by re-radiation from oxygen or nitrogen atoms which have been excited by collisions with these charged particles. 

Which effects cause radiation at which wavelengths/frequencies, and what actually causes the visible aurorae?
When the solar wind is funneled into the Earth's magnetic poles, those particles excite the electrons of molecules in the atmosphere which then bumps those electrons up into another orbital. When the electrons fall back down into their native orbital, they produce a photon of a particular wavelength whose energy is equal to the difference between the energy states of the two orbitals.There is a good pdf from NASA here and another good article here.