How does cold fusion work? I understand that in regular fusion, you use high levels of heat to accelerate atoms to speeds high enough to overcome nuclear force.  What method would be used to achieve cold fusion?
 A: When cold fusion hit the physics news in 1989 all the physicists in my institute followed it avidly and some tried to reproduce it. I have been following the news of cold fusion trying to keep an open mind after the disappointment of the next years.
The basic idea of all the experimental setups I have read about is to use the crystal structure of specific solids . In crystals the nuclei and electrons can have collective wave functions  that have probabilities of electrons and nuclei to exist all over the lattice. The hope of what is now called "low energy nuclear physics, LENR"  is that the coulomb barrier that will not allow a nucleus close to a nucleus can be thus defeated collectively, and the nuclei get close enough to produce fusion under certain conditions. 
The problems have been  that the experiments can not be consistently reproduced, and the attention of the physics community has been removed because of this disappointment. On the other hand a number of commercial enterprises have sprung up  trying to create units of cold fusion .  When/if  If/when a commercial unit appears the specific physical model  will not be long in appearing.
A: Basically the method to make cold fusion work is the same as hot fusion: "you use high levels of heat to accelerate atoms to speeds high enough to overcome nuclear force". 
You can make cold fusion work as follows. If you strip the electron off hydrogen atom, and let the proton capture a muon instead, then you have particle identical to the electron (as far as the Dirac equation description is concerned) aside from being 207 times heavier than the elctron and described by the Dirac equation with a central inverse square potential. So all the orbitals are the same shape and the whole thing looks exactly like a neutral Hydrogen atom aside from being 207 times smaller (the Bohr radius normalising the solution co-ordinates shrinks). Actually, the first quantised Dirac equation is getting a bit inaccurate as the muon is not so negligible in mass as the electron compared to the proton, and so you really have to do this problem as a quantum two-body problem and account for the non-fixed proton. But the basic idea stays the same: we've radically shrunken the "atom".
So now, with the proton's electric charge shielded by the much tighter muon orbital, it is found that room temperature kinetic energies are enough to overcome the coulombic repulsion, and cold fusion happens.
The only catch is that we need to have a source of muons and a factory for making shrunken hydrogen and this expends a great deal more energy than we get back from fusion. But I understand that putatively workable (and so far not successful) cold fusion schemes use roughly the same idea: the idea is to catalyse the reaction by getting the hydrogen's electron orbital to shrink somehow by combining it with something else and thus allow smaller energies in overcoming Coulombic repulsion. As dmckee points out, user Ron Maimon has a theory:

RonM has a theory. I don't have the background to fully evaluate it, so I can't say for sure if it is worth considering. My big question about it is: does it give a hint why it is hard to reliably build cells that work and stay working? (Or has someone solved that already?) 

I have read, although don't ask for details, that a tiny amount of cold fusion happens when peeling stickytape. It's simply that the number of events is too tiny to be of any energy production good.
